Theorem bj-2inf 13136

Description: Two formulations of the axiom of infinity (see ax-infvn 13139 and bj-omex 13140) . (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 2139	. . . 4 ⊢ ω = ω
2		bj-om 13135	. . . 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 13127	. . . . 5 ⊢ (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5		sseq1 3120	. . . . . . 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 1796	. . . . 5 ⊢ (𝑥 = ω → (∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
8	4, 7	anbi12d 464	. . . 4 ⊢ (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
9	8	spcegv 2774	. . 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 2689	. . . . . 6 ⊢ 𝑥 ∈ V
12		bj-om 13135	. . . . . 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 1579	. . 3 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ∃𝑥 𝑥 = ω)
16		isset 2692	. . 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 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686   ⊆ wss 3071  ωcom 4504  Ind wind 13124

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13011  ax-bdor 13014  ax-bdex 13017  ax-bdeq 13018  ax-bdel 13019  ax-bdsb 13020  ax-bdsep 13082

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13039  df-bj-ind 13125

This theorem is referenced by:  bj-omex  13140