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Mirrors > Home > MPE Home > Th. List > eqrel | Structured version Visualization version GIF version |
Description: Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) |
Ref | Expression |
---|---|
eqrel | ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrel 5795 | . . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
2 | ssrel 5795 | . . 3 ⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐵 → 〈𝑥, 𝑦〉 ∈ 𝐴))) | |
3 | 1, 2 | bi2anan9 638 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵) ∧ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐵 → 〈𝑥, 𝑦〉 ∈ 𝐴)))) |
4 | eqss 4011 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | 2albiim 1888 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵) ∧ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐵 → 〈𝑥, 𝑦〉 ∈ 𝐴))) | |
6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 〈cop 4637 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-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: eqrelriv 5802 eqrelrdv 5805 eqbrrdv 5806 eqrelrdv2 5808 opabid2 5841 reldm0 5941 iss 6055 asymref 6139 funssres 6612 fsn 7155 eqrelf 38237 iss2 38326 |
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