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| Mirrors > Home > MPE Home > Th. List > Mathboxes > epnsymrel | Structured version Visualization version GIF version | ||
| Description: The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.) |
| Ref | Expression |
|---|---|
| epnsymrel | ⊢ ¬ SymRel E |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | epnsym 9649 | . . . 4 ⊢ ◡ E ≠ E | |
| 2 | 1 | neii 2942 | . . 3 ⊢ ¬ ◡ E = E |
| 3 | 2 | intnanr 487 | . 2 ⊢ ¬ (◡ E = E ∧ Rel E ) |
| 4 | dfsymrel4 38552 | . 2 ⊢ ( SymRel E ↔ (◡ E = E ∧ Rel E )) | |
| 5 | 3, 4 | mtbir 323 | 1 ⊢ ¬ SymRel E |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 E cep 5583 ◡ccnv 5684 Rel wrel 5690 SymRel wsymrel 38194 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-fr 5637 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-symrel 38545 |
| This theorem is referenced by: (None) |
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