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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 9632 | . 2 ⊢ (◡ E ∩ E ) = ∅ | |
2 | disjeq0 4456 | . 2 ⊢ ((◡ E ∩ E ) = ∅ → (◡ E = E ↔ (◡ E = ∅ ∧ E = ∅))) | |
3 | epn0 5587 | . . . . . 6 ⊢ E ≠ ∅ | |
4 | eqneqall 2948 | . . . . . 6 ⊢ ( E = ∅ → ( E ≠ ∅ → ◡ E ≠ E )) | |
5 | 3, 4 | mpi 20 | . . . . 5 ⊢ ( E = ∅ → ◡ E ≠ E ) |
6 | 5 | adantl 481 | . . . 4 ⊢ ((◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E ) |
7 | 6 | a1i 11 | . . 3 ⊢ (◡ E = E → ((◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E )) |
8 | neqne 2945 | . . . 4 ⊢ (¬ ◡ E = E → ◡ E ≠ E ) | |
9 | 8 | a1d 25 | . . 3 ⊢ (¬ ◡ E = E → (¬ (◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E )) |
10 | 7, 9 | bija 380 | . 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 205 ∧ wa 395 = wceq 1534 ≠ wne 2937 ∩ cin 3946 ∅c0 4323 E cep 5581 ◡ccnv 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-reg 9616 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-eprel 5582 df-fr 5633 df-xp 5684 df-rel 5685 df-cnv 5686 |
This theorem is referenced by: epnsymrel 38034 |
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