Theorem eqelsuc 2307

Description: A set belongs to the successor of an equal set.

Hypothesis

Ref	Expression
eqelsuc.1	⊢ A ∈ V

Assertion

Ref	Expression
eqelsuc	⊢ (A = B → A ∈ suc B)

/TR ALIGN=LEFT>

Proof of Theorem eqelsuc

Step	Hyp	Ref	Expression
1		eqelsuc.1	. . 3 ⊢ A ∈ V
2	1	sucid 2304	. 2 ⊢ A ∈ suc A
3		suceq 2288	. . 3 ⊢ (A = B → suc A = suc B)
4	3	eleq2d 1156	. 2 ⊢ (A = B → (A ∈ suc A ↔ A ∈ suc B))
5	2, 4	mpbii 168	1 ⊢ (A = B → A ∈ suc B)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348  suc csuc 2201

This theorem is referenced by:  tfrlem11 2959  pssnn 3428

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-suc 2205

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