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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 6302 | . 2 ⊢ ¬ {{∅}} ∈ On | |
2 | snex 5322 | . . . 4 ⊢ {{∅}} ∈ V | |
3 | id 22 | . . . 4 ⊢ (On = V → On = V) | |
4 | 2, 3 | eleqtrrid 2917 | . . 3 ⊢ (On = V → {{∅}} ∈ On) |
5 | 4 | necon3bi 3039 | . 2 ⊢ (¬ {{∅}} ∈ On → On ≠ V) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ On ≠ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∅c0 4288 {csn 4557 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: (None) |
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