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Mirrors > Home > MPE Home > Th. List > onnevOLD | Structured version Visualization version GIF version |
Description: Obsolete version of onnev 6485 as of 27-May-2024. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
onnevOLD | ⊢ On ≠ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snsn0non 6483 | . 2 ⊢ ¬ {{∅}} ∈ On | |
2 | snex 5424 | . . . 4 ⊢ {{∅}} ∈ V | |
3 | id 22 | . . . 4 ⊢ (On = V → On = V) | |
4 | 2, 3 | eleqtrrid 2834 | . . 3 ⊢ (On = V → {{∅}} ∈ On) |
5 | 4 | necon3bi 2961 | . 2 ⊢ (¬ {{∅}} ∈ On → On ≠ V) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ On ≠ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 ∅c0 4317 {csn 4623 Oncon0 6358 |
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-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6361 df-on 6362 |
This theorem is referenced by: (None) |
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