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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 5031 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
5 | df-br 5031 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
6 | 3, 4, 5 | 3bitr3i 304 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) |
7 | 1, 2, 6 | eqrelriiv 5627 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: resco 6070 tpostpos 7895 sbthcl 8623 dfle2 12528 dflt2 12529 idsset 33464 dfbigcup2 33473 imageval 33504 inxpxrn 35803 |
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