Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7591 | . . . 4 ⊢ (ω ∈ On ∨ ω = On) | |
2 | 1 | ori 857 | . . 3 ⊢ (¬ ω ∈ On → ω = On) |
3 | onprc 7499 | . . . 4 ⊢ ¬ On ∈ V | |
4 | eleq1 2900 | . . . 4 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V)) | |
5 | 3, 4 | mtbiri 329 | . . 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 1537 ∈ wcel 2114 Vcvv 3494 Oncon0 6191 ωcom 7580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-om 7581 |
This theorem is referenced by: oaabs 8271 omelon 9109 fictb 9667 axdc3lem 9872 |
Copyright terms: Public domain | W3C validator |