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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 9600 | . 2 ⊢ (◡ E ∩ E ) = ∅ | |
2 | disjeq0 4448 | . 2 ⊢ ((◡ E ∩ E ) = ∅ → (◡ E = E ↔ (◡ E = ∅ ∧ E = ∅))) | |
3 | epn0 5576 | . . . . . 6 ⊢ E ≠ ∅ | |
4 | eqneqall 2943 | . . . . . 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 2940 | . . . 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 1533 ≠ wne 2932 ∩ cin 3940 ∅c0 4315 E cep 5570 ◡ccnv 5666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-reg 9584 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-eprel 5571 df-fr 5622 df-xp 5673 df-rel 5674 df-cnv 5675 |
This theorem is referenced by: epnsymrel 37936 |
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