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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 7898 | . . . 4 ⊢ (ω ∈ On ∨ ω = On) | |
2 | 1 | ori 861 | . . 3 ⊢ (¬ ω ∈ On → ω = On) |
3 | onprc 7796 | . . . 4 ⊢ ¬ On ∈ V | |
4 | eleq1 2826 | . . . 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 1536 ∈ wcel 2105 Vcvv 3477 Oncon0 6385 ωcom 7886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 df-lim 6390 df-om 7887 |
This theorem is referenced by: oaabs 8684 omelon 9683 fictb 10281 axdc3lem 10487 |
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