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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 7118 | . . . 4 ⊢ (ω ∈ On ∨ ω = On) | |
2 | 1 | ori 389 | . . 3 ⊢ (¬ ω ∈ On → ω = On) |
3 | onprc 7026 | . . . 4 ⊢ ¬ On ∈ V | |
4 | eleq1 2718 | . . . 4 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V)) | |
5 | 3, 4 | mtbiri 316 | . . 3 ⊢ (ω = On → ¬ ω ∈ V) |
6 | 2, 5 | syl 17 | . 2 ⊢ (¬ ω ∈ On → ¬ ω ∈ V) |
7 | 6 | con4i 113 | 1 ⊢ (ω ∈ V → ω ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 Oncon0 5761 ωcom 7107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-tr 4786 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-om 7108 |
This theorem is referenced by: oaabs 7769 omelon 8581 fictb 9105 axdc3lem 9310 |
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