Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
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 13066 (see bj-2inf 13063 for the equivalence of the latter with bj-omex 13067). (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 13101 | . . 3 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) | |
2 | vex 2663 | . . . 4 ⊢ 𝑎 ∈ V | |
3 | bdcv 12973 | . . . . 5 ⊢ BOUNDED 𝑎 | |
4 | 3 | bj-inf2vn 13099 | . . . 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 1566 | . 2 ⊢ ∃𝑎 𝑎 = ω |
7 | 6 | issetri 2669 | 1 ⊢ ω ∈ V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 682 ∀wal 1314 = wceq 1316 ∈ wcel 1465 ∃wrex 2394 Vcvv 2660 ∅c0 3333 suc csuc 4257 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-bdim 12939 ax-bdor 12941 ax-bdex 12944 ax-bdeq 12945 ax-bdel 12946 ax-bdsb 12947 ax-bdsep 13009 ax-bdsetind 13093 ax-inf2 13101 |
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-suc 4263 df-iom 4475 df-bdc 12966 df-bj-ind 13052 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |