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Mirrors > Home > MPE Home > Th. List > eqbrriv | Structured version Visualization version GIF version |
Description: Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.) |
Ref | Expression |
---|---|
eqbrriv.1 | ⊢ Rel 𝐴 |
eqbrriv.2 | ⊢ Rel 𝐵 |
eqbrriv.3 | ⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦) |
Ref | Expression |
---|---|
eqbrriv | ⊢ 𝐴 = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrriv.1 | . 2 ⊢ Rel 𝐴 | |
2 | eqbrriv.2 | . 2 ⊢ Rel 𝐵 | |
3 | eqbrriv.3 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦) | |
4 | df-br 5111 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
5 | df-br 5111 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) | |
6 | 3, 4, 5 | 3bitr3i 301 | . 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) |
7 | 1, 2, 6 | eqrelriiv 5751 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ⟨cop 4597 class class class wbr 5110 Rel wrel 5643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-in 3922 df-ss 3932 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 |
This theorem is referenced by: resco 6207 tpostpos 8182 sbthcl 9046 dfle2 13073 dflt2 13074 xpab 34337 idsset 34504 dfbigcup2 34513 imageval 34544 inxpxrn 36886 |
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