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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-omex | GIF version |
Description: Proof of omex 4316 from ax-infvn 10066. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-omex | ⊢ ω ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-infvn 10066 | . 2 ⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) | |
2 | bj-2inf 10062 | . 2 ⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) | |
3 | 1, 2 | mpbir 134 | 1 ⊢ ω ∈ V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 ⊆ wss 2917 ωcom 4313 Ind wind 10050 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 ax-pr 3944 ax-un 4170 ax-bd0 9933 ax-bdor 9936 ax-bdex 9939 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 ax-bdsep 10004 ax-infvn 10066 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 df-bdc 9961 df-bj-ind 10051 |
This theorem is referenced by: bdpeano5 10068 speano5 10069 bdfind 10071 bj-omtrans 10081 bj-omelon 10086 |
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