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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 5144 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
| 5 | df-br 5144 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
| 6 | 3, 4, 5 | 3bitr3i 301 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) |
| 7 | 1, 2, 6 | eqrelriiv 5800 | 1 ⊢ 𝐴 = 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: resco 6270 tpostpos 8271 sbthcl 9135 dfle2 13189 dflt2 13190 xpab 35726 idsset 35891 dfbigcup2 35900 imageval 35931 inxpxrn 38396 tposres0 48777 |
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