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Mirrors > Home > MPE Home > Th. List > onnev | Structured version Visualization version GIF version |
Description: The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) (Proof shortened by Wolf Lammen, 27-May-2024.) |
Ref | Expression |
---|---|
onnev | ⊢ On ≠ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snsn0non 6488 | . . 3 ⊢ ¬ {{∅}} ∈ On | |
2 | snex 5427 | . . . 4 ⊢ {{∅}} ∈ V | |
3 | id 22 | . . . 4 ⊢ (On = V → On = V) | |
4 | 2, 3 | eleqtrrid 2835 | . . 3 ⊢ (On = V → {{∅}} ∈ On) |
5 | 1, 4 | mto 196 | . 2 ⊢ ¬ On = V |
6 | 5 | neir 2938 | 1 ⊢ On ≠ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ≠ wne 2935 Vcvv 3469 ∅c0 4318 {csn 4624 Oncon0 6363 |
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-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6366 df-on 6367 |
This theorem is referenced by: (None) |
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