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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-omord | GIF version |
Description: The set ω is an ordinal class. Constructive proof of ordom 4618. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-omord | ⊢ Ord ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-omtrans2 15062 | . 2 ⊢ Tr ω | |
2 | bj-nntrans2 15057 | . . 3 ⊢ (𝑥 ∈ ω → Tr 𝑥) | |
3 | 2 | rgen 2540 | . 2 ⊢ ∀𝑥 ∈ ω Tr 𝑥 |
4 | dford3 4379 | . 2 ⊢ (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥)) | |
5 | 1, 3, 4 | mpbir2an 943 | 1 ⊢ Ord ω |
Colors of variables: wff set class |
Syntax hints: ∀wral 2465 Tr wtr 4113 Ord word 4374 ωcom 4601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-nul 4141 ax-pr 4221 ax-un 4445 ax-bd0 14918 ax-bdor 14921 ax-bdal 14923 ax-bdex 14924 ax-bdeq 14925 ax-bdel 14926 ax-bdsb 14927 ax-bdsep 14989 ax-infvn 15046 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-sn 3610 df-pr 3611 df-uni 3822 df-int 3857 df-tr 4114 df-iord 4378 df-suc 4383 df-iom 4602 df-bdc 14946 df-bj-ind 15032 |
This theorem is referenced by: bj-omelon 15066 |
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