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Mirrors > Home > MPE Home > Th. List > eqrelriv | Structured version Visualization version GIF version |
Description: Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.) |
Ref | Expression |
---|---|
eqrelriv.1 | ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) |
Ref | Expression |
---|---|
eqrelriv | ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrelriv.1 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
2 | 1 | gen2 1798 | . 2 ⊢ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) |
3 | eqrel 5738 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
4 | 2, 3 | mpbiri 257 | 1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 〈cop 4590 Rel wrel 5636 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3445 df-in 3915 df-ss 3925 df-opab 5166 df-xp 5637 df-rel 5638 |
This theorem is referenced by: eqrelriiv 5744 dfrel2 6139 coi1 6212 cnviin 6236 |
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