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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 5142 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
5 | df-br 5142 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) | |
6 | 3, 4, 5 | 3bitr3i 301 | . 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) |
7 | 1, 2, 6 | eqrelriiv 5783 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ⟨cop 4629 class class class wbr 5141 Rel wrel 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-in 3950 df-ss 3960 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 |
This theorem is referenced by: resco 6242 tpostpos 8229 sbthcl 9094 dfle2 13129 dflt2 13130 xpab 35229 idsset 35395 dfbigcup2 35404 imageval 35435 inxpxrn 37776 |
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