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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-omelon | GIF version |
Description: The set ω is an ordinal. Constructive proof of omelon 4349. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-omelon | ⊢ ω ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-omord 10755 | . 2 ⊢ Ord ω | |
2 | bj-omex 10737 | . . 3 ⊢ ω ∈ V | |
3 | 2 | elon 4129 | . 2 ⊢ (ω ∈ On ↔ Ord ω) |
4 | 1, 3 | mpbir 144 | 1 ⊢ ω ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 Ord word 4117 Oncon0 4118 ωcom 4331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 ax-pr 3964 ax-un 4188 ax-bd0 10604 ax-bdor 10607 ax-bdal 10609 ax-bdex 10610 ax-bdeq 10611 ax-bdel 10612 ax-bdsb 10613 ax-bdsep 10675 ax-infvn 10736 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-bdc 10632 df-bj-ind 10722 |
This theorem is referenced by: bj-omssonALT 10758 |
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