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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 9073 | . 2 ⊢ (◡ E ∩ E ) = ∅ | |
2 | disjeq0 4407 | . 2 ⊢ ((◡ E ∩ E ) = ∅ → (◡ E = E ↔ (◡ E = ∅ ∧ E = ∅))) | |
3 | epn0 5473 | . . . . . 6 ⊢ E ≠ ∅ | |
4 | eqneqall 3029 | . . . . . 6 ⊢ ( E = ∅ → ( E ≠ ∅ → ◡ E ≠ E )) | |
5 | 3, 4 | mpi 20 | . . . . 5 ⊢ ( E = ∅ → ◡ E ≠ E ) |
6 | 5 | adantl 484 | . . . 4 ⊢ ((◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E ) |
7 | 6 | a1i 11 | . . 3 ⊢ (◡ E = E → ((◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E )) |
8 | neqne 3026 | . . . 4 ⊢ (¬ ◡ E = E → ◡ E ≠ E ) | |
9 | 8 | a1d 25 | . . 3 ⊢ (¬ ◡ E = E → (¬ (◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E )) |
10 | 7, 9 | bija 384 | . 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 208 ∧ wa 398 = wceq 1537 ≠ wne 3018 ∩ cin 3937 ∅c0 4293 E cep 5466 ◡ccnv 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-reg 9058 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-eprel 5467 df-fr 5516 df-xp 5563 df-rel 5564 df-cnv 5565 |
This theorem is referenced by: epnsymrel 35800 |
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