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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 8788 | . . . 4 ⊢ ◡ E ≠ E | |
2 | 1 | neii 3001 | . . 3 ⊢ ¬ ◡ E = E |
3 | 2 | intnanr 483 | . 2 ⊢ ¬ (◡ E = E ∧ Rel E ) |
4 | dfsymrel4 34840 | . 2 ⊢ ( SymRel E ↔ (◡ E = E ∧ Rel E )) | |
5 | 3, 4 | mtbir 315 | 1 ⊢ ¬ SymRel E |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 386 = wceq 1656 E cep 5256 ◡ccnv 5345 Rel wrel 5351 SymRel wsymrel 34531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-reg 8773 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-eprel 5257 df-fr 5305 df-xp 5352 df-rel 5353 df-cnv 5354 df-dm 5356 df-rn 5357 df-res 5358 df-symrel 34833 |
This theorem is referenced by: (None) |
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