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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 5621 | . . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
2 | ssrel 5621 | . . 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 3930 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | 2albiim 1891 | . 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 317 | 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 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 〈cop 4531 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-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: eqrelriv 5626 eqrelrdv 5629 eqbrrdv 5630 eqrelrdv2 5632 opabid2 5664 reldm0 5762 iss 5870 asymref 5943 funssres 6368 fsn 6874 eqrelf 35677 iss2 35761 |
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