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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-omex2 | GIF version |
Description: Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 13310 (see bj-2inf 13307 for the equivalence of the latter with bj-omex 13311). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-omex2 | ⊢ ω ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-inf2 13345 | . . 3 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) | |
2 | vex 2692 | . . . 4 ⊢ 𝑎 ∈ V | |
3 | bdcv 13217 | . . . . 5 ⊢ BOUNDED 𝑎 | |
4 | 3 | bj-inf2vn 13343 | . . . 4 ⊢ (𝑎 ∈ V → (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω)) |
5 | 2, 4 | ax-mp 5 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) → 𝑎 = ω) |
6 | 1, 5 | eximii 1582 | . 2 ⊢ ∃𝑎 𝑎 = ω |
7 | 6 | issetri 2698 | 1 ⊢ ω ∈ V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 698 ∀wal 1330 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 Vcvv 2689 ∅c0 3368 suc csuc 4295 ωcom 4512 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-nul 4062 ax-pr 4139 ax-un 4363 ax-bd0 13182 ax-bdim 13183 ax-bdor 13185 ax-bdex 13188 ax-bdeq 13189 ax-bdel 13190 ax-bdsb 13191 ax-bdsep 13253 ax-bdsetind 13337 ax-inf2 13345 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-suc 4301 df-iom 4513 df-bdc 13210 df-bj-ind 13296 |
This theorem is referenced by: (None) |
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