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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-omex | GIF version |
Description: Proof of omex 4589 from ax-infvn 14346. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-omex | ⊢ ω ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-infvn 14346 | . 2 ⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) | |
2 | bj-2inf 14343 | . 2 ⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) | |
3 | 1, 2 | mpbir 146 | 1 ⊢ ω ∈ V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1351 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 ωcom 4586 Ind wind 14331 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-nul 4126 ax-pr 4206 ax-un 4430 ax-bd0 14218 ax-bdor 14221 ax-bdex 14224 ax-bdeq 14225 ax-bdel 14226 ax-bdsb 14227 ax-bdsep 14289 ax-infvn 14346 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-sn 3597 df-pr 3598 df-uni 3808 df-int 3843 df-suc 4368 df-iom 4587 df-bdc 14246 df-bj-ind 14332 |
This theorem is referenced by: bdpeano5 14348 speano5 14349 bdfind 14351 bj-omtrans 14361 bj-omelon 14366 |
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