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Mirrors > Home > MPE Home > Th. List > omelon2 | Structured version Visualization version GIF version |
Description: Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
Ref | Expression |
---|---|
omelon2 | ⊢ (ω ∈ V → ω ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omon 7861 | . . . 4 ⊢ (ω ∈ On ∨ ω = On) | |
2 | 1 | ori 858 | . . 3 ⊢ (¬ ω ∈ On → ω = On) |
3 | onprc 7759 | . . . 4 ⊢ ¬ On ∈ V | |
4 | eleq1 2813 | . . . 4 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V)) | |
5 | 3, 4 | mtbiri 327 | . . 3 ⊢ (ω = On → ¬ ω ∈ V) |
6 | 2, 5 | syl 17 | . 2 ⊢ (¬ ω ∈ On → ¬ ω ∈ V) |
7 | 6 | con4i 114 | 1 ⊢ (ω ∈ V → ω ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3466 Oncon0 6355 ωcom 7849 |
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 df-lim 6360 df-om 7850 |
This theorem is referenced by: oaabs 8644 omelon 9638 fictb 10237 axdc3lem 10442 |
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