Theorem cleqreli 2484

Description: Inference from extensionality principle for relations.

Hypotheses

Ref	Expression
cleqreli.1	⊢ Rel A
cleqreli.2	⊢ Rel B
cleqreli.3	⊢ (⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)

Assertion

Ref	Expression
cleqreli	⊢ A = B

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem cleqreli

Step	Hyp	Ref	Expression
1		cleqreli.1	. . 3 ⊢ Rel A
2		cleqreli.2	. . 3 ⊢ Rel B
3		cleqrel 2483	. . 3 ⊢ ((Rel A ∧ Rel B) → (A = B ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)))
4	1, 2, 3	mp2an 520	. 2 ⊢ (A = B ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B))
5		cleqreli.3	. . 3 ⊢ (⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)
6	5	ax-gen 677	. 2 ⊢ ∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)
7	4, 6	mpgbir 686	1 ⊢ A = B

Syntax hints:   ↔ wb 127  ∀wal 672   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  Rel wrel 2415

This theorem is referenced by:  inopab 2495  inxp 2496  cnvopab 2632  cnv0 2633  cnvi 2634  cnvsn 2636  cnvun 2642  cnvin 2643  cnvxp 2651  co02 2663  coass 2667  sbthcl 3361

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425

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