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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 9552 | . 2 ⊢ (◡ E ∩ E ) = ∅ | |
2 | disjeq0 4419 | . 2 ⊢ ((◡ E ∩ E ) = ∅ → (◡ E = E ↔ (◡ E = ∅ ∧ E = ∅))) | |
3 | epn0 5546 | . . . . . 6 ⊢ E ≠ ∅ | |
4 | eqneqall 2951 | . . . . . 6 ⊢ ( E = ∅ → ( E ≠ ∅ → ◡ E ≠ E )) | |
5 | 3, 4 | mpi 20 | . . . . 5 ⊢ ( E = ∅ → ◡ E ≠ E ) |
6 | 5 | adantl 483 | . . . 4 ⊢ ((◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E ) |
7 | 6 | a1i 11 | . . 3 ⊢ (◡ E = E → ((◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E )) |
8 | neqne 2948 | . . . 4 ⊢ (¬ ◡ E = E → ◡ E ≠ E ) | |
9 | 8 | a1d 25 | . . 3 ⊢ (¬ ◡ E = E → (¬ (◡ E = ∅ ∧ E = ∅) → ◡ E ≠ E )) |
10 | 7, 9 | bija 382 | . 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 397 = wceq 1542 ≠ wne 2940 ∩ cin 3913 ∅c0 4286 E cep 5540 ◡ccnv 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-reg 9536 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-eprel 5541 df-fr 5592 df-xp 5643 df-rel 5644 df-cnv 5645 |
This theorem is referenced by: epnsymrel 37074 |
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