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Mirrors > Home > MPE Home > Th. List > Mathboxes > antisymrelressn | Structured version Visualization version GIF version |
Description: (Contributed by Peter Mazsa, 29-Jun-2024.) |
Ref | Expression |
---|---|
antisymrelressn | ⊢ AntisymRel (𝑅 ↾ {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | antisymressn 36837 | . 2 ⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) | |
2 | relres 5964 | . 2 ⊢ Rel (𝑅 ↾ {𝐴}) | |
3 | dfantisymrel5 37155 | . 2 ⊢ ( AntisymRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) ∧ Rel (𝑅 ↾ {𝐴}))) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ AntisymRel (𝑅 ↾ {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1539 = wceq 1541 {csn 4584 class class class wbr 5103 ↾ cres 5633 Rel wrel 5636 AntisymRel wantisymrel 36602 |
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-10 2137 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-dm 5641 df-rn 5642 df-res 5643 df-cnvrefrel 36920 df-antisymrel 37153 |
This theorem is referenced by: (None) |
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