Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6310 | . . 3 ⊢ ¬ {{∅}} ∈ On | |
2 | snex 5309 | . . . 4 ⊢ {{∅}} ∈ V | |
3 | id 22 | . . . 4 ⊢ (On = V → On = V) | |
4 | 2, 3 | eleqtrrid 2838 | . . 3 ⊢ (On = V → {{∅}} ∈ On) |
5 | 1, 4 | mto 200 | . 2 ⊢ ¬ On = V |
6 | 5 | neir 2935 | 1 ⊢ On ≠ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ∅c0 4223 {csn 4527 Oncon0 6191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |