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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.) |
Ref | Expression |
---|---|
onnev | ⊢ On ≠ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snsn0non 6082 | . 2 ⊢ ¬ {{∅}} ∈ On | |
2 | snex 5130 | . . . 4 ⊢ {{∅}} ∈ V | |
3 | id 22 | . . . 4 ⊢ (On = V → On = V) | |
4 | 2, 3 | syl5eleqr 2914 | . . 3 ⊢ (On = V → {{∅}} ∈ On) |
5 | 4 | necon3bi 3026 | . 2 ⊢ (¬ {{∅}} ∈ On → On ≠ V) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ On ≠ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 Vcvv 3415 ∅c0 4145 {csn 4398 Oncon0 5964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-tr 4977 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-ord 5967 df-on 5968 |
This theorem is referenced by: (None) |
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