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Mirrors > Home > MPE Home > Th. List > epnsym | Structured version Visualization version GIF version |
Description: The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.) |
Ref | Expression |
---|---|
epnsym | ⊢ ◡ E ≠ E |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvepnep 8800 | . 2 ⊢ (◡ E ∩ E ) = ∅ | |
2 | disjeq0 4248 | . 2 ⊢ ((◡ E ∩ E ) = ∅ → (◡ E = E ↔ (◡ E = ∅ ∧ E = ∅))) | |
3 | epn0 5271 | . . . . . 6 ⊢ E ≠ ∅ | |
4 | eqneqall 2980 | . . . . . 6 ⊢ ( E = ∅ → ( E ≠ ∅ → ◡ E ≠ E )) | |
5 | 3, 4 | mpi 20 | . . . . 5 ⊢ ( E = ∅ → ◡ E ≠ E ) |
6 | 5 | adantl 475 | . . . 4 ⊢ ((◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E ) |
7 | 6 | a1i 11 | . . 3 ⊢ (◡ E = E → ((◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E )) |
8 | neqne 2977 | . . . 4 ⊢ (¬ ◡ E = E → ◡ E ≠ E ) | |
9 | 8 | a1d 25 | . . 3 ⊢ (¬ ◡ E = E → (¬ (◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E )) |
10 | 7, 9 | bija 372 | . 2 ⊢ ((◡ E = E ↔ (◡ E = ∅ ∧ E = ∅)) → ◡ E ≠ E ) |
11 | 1, 2, 10 | mp2b 10 | 1 ⊢ ◡ E ≠ E |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ≠ wne 2969 ∩ cin 3791 ∅c0 4141 E cep 5265 ◡ccnv 5354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-reg 8786 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-eprel 5266 df-fr 5314 df-xp 5361 df-rel 5362 df-cnv 5363 |
This theorem is referenced by: epnsymrel 34936 |
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