Theorem bj-2inf 13820

Description: Two formulations of the axiom of infinity (see ax-infvn 13823 and bj-omex 13824) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-2inf	⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-2inf

Step	Hyp	Ref	Expression
1		eqid 2165	. . . 4 ⊢ ω = ω
2		bj-om 13819	. . . 4 ⊢ (ω ∈ V → (ω = ω ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
3	1, 2	mpbii 147	. . 3 ⊢ (ω ∈ V → (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
4		bj-indeq 13811	. . . . 5 ⊢ (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5		sseq1 3165	. . . . . . 7 ⊢ (𝑥 = ω → (𝑥 ⊆ 𝑦 ↔ ω ⊆ 𝑦))
6	5	imbi2d 229	. . . . . 6 ⊢ (𝑥 = ω → ((Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦)))
7	6	albidv 1812	. . . . 5 ⊢ (𝑥 = ω → (∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
8	4, 7	anbi12d 465	. . . 4 ⊢ (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
9	8	spcegv 2814	. . 3 ⊢ (ω ∈ V → ((Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)) → ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))))
10	3, 9	mpd 13	. 2 ⊢ (ω ∈ V → ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))
11		vex 2729	. . . . . 6 ⊢ 𝑥 ∈ V
12		bj-om 13819	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))))
13	11, 12	ax-mp 5	. . . . 5 ⊢ (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))
14	13	biimpri 132	. . . 4 ⊢ ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → 𝑥 = ω)
15	14	eximi 1588	. . 3 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ∃𝑥 𝑥 = ω)
16		isset 2732	. . 3 ⊢ (ω ∈ V ↔ ∃𝑥 𝑥 = ω)
17	15, 16	sylibr 133	. 2 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ω ∈ V)
18	10, 17	impbii 125	1 ⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  ωcom 4567  Ind wind 13808

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-nul 4108  ax-pr 4187  ax-un 4411  ax-bd0 13695  ax-bdor 13698  ax-bdex 13701  ax-bdeq 13702  ax-bdel 13703  ax-bdsb 13704  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-suc 4349  df-iom 4568  df-bdc 13723  df-bj-ind 13809

This theorem is referenced by:  bj-omex  13824