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Mirrors > Home > MPE Home > Th. List > Mathboxes > antisymressn | Structured version Visualization version GIF version |
Description: Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37838) is antisymmetric. (Contributed by Peter Mazsa, 11-Jun-2024.) |
Ref | Expression |
---|---|
antisymressn | ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brressn 37837 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦))) | |
2 | 1 | el2v 3477 | . . . 4 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦)) |
3 | 2 | simplbi 497 | . . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 → 𝑥 = 𝐴) |
4 | brressn 37837 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥))) | |
5 | 4 | el2v 3477 | . . . 4 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥)) |
6 | 5 | simplbi 497 | . . 3 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 → 𝑦 = 𝐴) |
7 | eqtr3 2753 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦) | |
8 | 3, 6, 7 | syl2an 595 | . 2 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) |
9 | 8 | gen2 1791 | 1 ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 = wceq 1534 Vcvv 3469 {csn 4624 class class class wbr 5142 ↾ cres 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-res 5684 |
This theorem is referenced by: antisymrelressn 38160 |
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