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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 6480 | . . 3 ⊢ ¬ {{∅}} ∈ On | |
2 | snex 5422 | . . . 4 ⊢ {{∅}} ∈ V | |
3 | id 22 | . . . 4 ⊢ (On = V → On = V) | |
4 | 2, 3 | eleqtrrid 2832 | . . 3 ⊢ (On = V → {{∅}} ∈ On) |
5 | 1, 4 | mto 196 | . 2 ⊢ ¬ On = V |
6 | 5 | neir 2935 | 1 ⊢ On ≠ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ∅c0 4315 {csn 4621 Oncon0 6355 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
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 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-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-tr 5257 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-ord 6358 df-on 6359 |
This theorem is referenced by: (None) |
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