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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 9056 | . . . 4 ⊢ ◡ E ≠ E | |
2 | 1 | neii 2989 | . . 3 ⊢ ¬ ◡ E = E |
3 | 2 | intnanr 491 | . 2 ⊢ ¬ (◡ E = E ∧ Rel E ) |
4 | dfsymrel4 35947 | . 2 ⊢ ( SymRel E ↔ (◡ E = E ∧ Rel E )) | |
5 | 3, 4 | mtbir 326 | 1 ⊢ ¬ SymRel E |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 E cep 5429 ◡ccnv 5518 Rel wrel 5524 SymRel wsymrel 35625 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-reg 9040 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-fr 5478 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-symrel 35940 |
This theorem is referenced by: (None) |
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