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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 5149 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
5 | df-br 5149 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
6 | 3, 4, 5 | 3bitr3i 301 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) |
7 | 1, 2, 6 | eqrelriiv 5803 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: resco 6272 tpostpos 8270 sbthcl 9134 dfle2 13186 dflt2 13187 xpab 35706 idsset 35872 dfbigcup2 35881 imageval 35912 inxpxrn 38377 |
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