Theorem FstSum | index | src |

theorem FstSum (A B: set): $ Fst (Sum A B) == A $;
StepHypRefExpression
1 bitr
(a1 e. Fst (Sum A B) <-> b0 a1 e. Sum A B) -> (b0 a1 e. Sum A B <-> a1 e. A) -> (a1 e. Fst (Sum A B) <-> a1 e. A)
2 elFst
a1 e. Fst (Sum A B) <-> b0 a1 e. Sum A B
3 1, 2 ax_mp
(b0 a1 e. Sum A B <-> a1 e. A) -> (a1 e. Fst (Sum A B) <-> a1 e. A)
4 Suml
b0 a1 e. Sum A B <-> a1 e. A
5 3, 4 ax_mp
a1 e. Fst (Sum A B) <-> a1 e. A
6 5 ax_gen
A. a1 (a1 e. Fst (Sum A B) <-> a1 e. A)
7 6 conv eqs
Fst (Sum A B) == A

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)