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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-omord | GIF version |
Description: The set ω is an ordinal. Constructive proof of ordom 4490. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-omord | ⊢ Ord ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-omtrans2 13082 | . 2 ⊢ Tr ω | |
2 | bj-nntrans2 13077 | . . 3 ⊢ (𝑥 ∈ ω → Tr 𝑥) | |
3 | 2 | rgen 2462 | . 2 ⊢ ∀𝑥 ∈ ω Tr 𝑥 |
4 | dford3 4259 | . 2 ⊢ (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥)) | |
5 | 1, 3, 4 | mpbir2an 911 | 1 ⊢ Ord ω |
Colors of variables: wff set class |
Syntax hints: ∀wral 2393 Tr wtr 3996 Ord word 4254 ωcom 4474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-nul 4024 ax-pr 4101 ax-un 4325 ax-bd0 12938 ax-bdor 12941 ax-bdal 12943 ax-bdex 12944 ax-bdeq 12945 ax-bdel 12946 ax-bdsb 12947 ax-bdsep 13009 ax-infvn 13066 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-tr 3997 df-iord 4258 df-suc 4263 df-iom 4475 df-bdc 12966 df-bj-ind 13052 |
This theorem is referenced by: bj-omelon 13086 |
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