Theorem elins | index | src |

pub theorem elins (a b c: nat): $ a e. b ; c <-> a = b \/ a e. c $;
StepHypRefExpression
2
finite {x | x = b \/ x e. c} -> (a e. lower {x | x = b \/ x e. c} <-> a e. {x | x = b \/ x e. c})
3
conv ins
finite {x | x = b \/ x e. c} -> (a e. b ; c <-> a e. {x | x = b \/ x e. c})
5
A. x (x = b \/ x e. c -> x <= max b c) <-> {x | x = b \/ x e. c} C_ {x | x <= max b c}
7
b <= max b c
8
x = b -> (x <= max b c <-> b <= max b c)
9
7, 8
x = b -> x <= max b c
11
x e. c -> x < c
12
x < c -> x <= c
13
c <= max b c
14
x < c -> c <= max b c
15
x < c -> x <= max b c
16
x e. c -> x <= max b c
17
9, 16
x = b \/ x e. c -> x <= max b c
18
A. x (x = b \/ x e. c -> x <= max b c)
19
5, 18
{x | x = b \/ x e. c} C_ {x | x <= max b c}
21
finite {x | x <= max b c}
22
finite {x | x = b \/ x e. c}
23
3, 22
a e. b ; c <-> a e. {x | x = b \/ x e. c}
25
x = a -> (x = b <-> a = b)
26
x = a -> (x e. c <-> a e. c)
27
x = a -> (x = b \/ x e. c <-> a = b \/ a e. c)
28
a e. {x | x = b \/ x e. c} <-> a = b \/ a e. c
29
a e. b ; c <-> a = b \/ a e. c

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)