Theorem rneq0 ≪ | index | src | ≫ 

theorem rneq0 (A: set): $ Ran A == 0 <-> A == 0 $;
StepHypRefExpression
1 bitr3 
(Ran A C_ 0 <-> Ran A == 0) -> (Ran A C_ 0 <-> A == 0) -> (Ran A == 0 <-> A == 0)
2 ss02 
Ran A C_ 0 <-> Ran A == 0
3 1, 2 ax_mp 
(Ran A C_ 0 <-> A == 0) -> (Ran A == 0 <-> A == 0)
4 bitr 
(Ran A C_ 0 <-> A C_ Xp _V 0) -> (A C_ Xp _V 0 <-> A == 0) -> (Ran A C_ 0 <-> A == 0)
5 ssrn 
Ran A C_ 0 <-> A C_ Xp _V 0
6 4, 5 ax_mp 
(A C_ Xp _V 0 <-> A == 0) -> (Ran A C_ 0 <-> A == 0)
7 bitr 
(A C_ Xp _V 0 <-> A C_ 0) -> (A C_ 0 <-> A == 0) -> (A C_ Xp _V 0 <-> A == 0)
8 sseq2 
Xp _V 0 == 0 -> (A C_ Xp _V 0 <-> A C_ 0)
9 xp02 
Xp _V 0 == 0
10 8, 9 ax_mp 
A C_ Xp _V 0 <-> A C_ 0
11 7, 10 ax_mp 
(A C_ 0 <-> A == 0) -> (A C_ Xp _V 0 <-> A == 0)
12 ss02 
A C_ 0 <-> A == 0
13 11, 12 ax_mp 
A C_ Xp _V 0 <-> A == 0
14 6, 13 ax_mp 
Ran A C_ 0 <-> A == 0
15 3, 14 ax_mp 
Ran A == 0 <-> A == 0

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)