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Mirrors > Home > MPE Home > Th. List > onnevOLD | Structured version Visualization version GIF version |
Description: Obsolete version of onnev 6283 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 6281 | . 2 ⊢ ¬ {{∅}} ∈ On | |
2 | snex 5293 | . . . 4 ⊢ {{∅}} ∈ V | |
3 | id 22 | . . . 4 ⊢ (On = V → On = V) | |
4 | 2, 3 | eleqtrrid 2858 | . . 3 ⊢ (On = V → {{∅}} ∈ On) |
5 | 4 | necon3bi 2975 | . 2 ⊢ (¬ {{∅}} ∈ On → On ≠ V) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ On ≠ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2112 ≠ wne 2949 Vcvv 3407 ∅c0 4221 {csn 4515 Oncon0 6162 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-tr 5132 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-ord 6165 df-on 6166 |
This theorem is referenced by: (None) |
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