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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 37966) 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 37965 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦))) | |
2 | 1 | el2v 3471 | . . . 4 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦)) |
3 | 2 | simplbi 496 | . . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 → 𝑥 = 𝐴) |
4 | brressn 37965 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥))) | |
5 | 4 | el2v 3471 | . . . 4 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑥)) |
6 | 5 | simplbi 496 | . . 3 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑥 → 𝑦 = 𝐴) |
7 | eqtr3 2751 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦) | |
8 | 3, 6, 7 | syl2an 594 | . 2 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) |
9 | 8 | gen2 1790 | 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 394 ∀wal 1531 = wceq 1533 Vcvv 3463 {csn 4625 class class class wbr 5144 ↾ cres 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5145 df-opab 5207 df-xp 5679 df-res 5685 |
This theorem is referenced by: antisymrelressn 38288 |
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