Theorem bj-2inf 11174

Description: Two formulations of the axiom of infinity (see ax-infvn 11177 and bj-omex 11178) . (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 2083	. . . 4 ⊢ ω = ω
2		bj-om 11173	. . . 4 ⊢ (ω ∈ V → (ω = ω ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
3	1, 2	mpbii 146	. . 3 ⊢ (ω ∈ V → (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
4		bj-indeq 11165	. . . . 5 ⊢ (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5		sseq1 3031	. . . . . . 7 ⊢ (𝑥 = ω → (𝑥 ⊆ 𝑦 ↔ ω ⊆ 𝑦))
6	5	imbi2d 228	. . . . . 6 ⊢ (𝑥 = ω → ((Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦)))
7	6	albidv 1747	. . . . 5 ⊢ (𝑥 = ω → (∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
8	4, 7	anbi12d 457	. . . 4 ⊢ (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
9	8	spcegv 2697	. . 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 2615	. . . . . 6 ⊢ 𝑥 ∈ V
12		bj-om 11173	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))))
13	11, 12	ax-mp 7	. . . . 5 ⊢ (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))
14	13	biimpri 131	. . . 4 ⊢ ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → 𝑥 = ω)
15	14	eximi 1532	. . 3 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ∃𝑥 𝑥 = ω)
16		isset 2616	. . 3 ⊢ (ω ∈ V ↔ ∃𝑥 𝑥 = ω)
17	15, 16	sylibr 132	. 2 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ω ∈ V)
18	10, 17	impbii 124	1 ⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1283   = wceq 1285  ∃wex 1422   ∈ wcel 1434  Vcvv 2612   ⊆ wss 2984  ωcom 4367  Ind wind 11162

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3930  ax-pr 3999  ax-un 4223  ax-bd0 11045  ax-bdor 11048  ax-bdex 11051  ax-bdeq 11052  ax-bdel 11053  ax-bdsb 11054  ax-bdsep 11116

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-suc 4161  df-iom 4368  df-bdc 11073  df-bj-ind 11163

This theorem is referenced by:  bj-omex  11178