Theorem bj-2inf 13973

Description: Two formulations of the axiom of infinity (see ax-infvn 13976 and bj-omex 13977) . (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 2170	. . . 4 ⊢ ω = ω
2		bj-om 13972	. . . 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 13964	. . . . 5 ⊢ (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5		sseq1 3170	. . . . . . 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 1817	. . . . 5 ⊢ (𝑥 = ω → (∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
8	4, 7	anbi12d 470	. . . 4 ⊢ (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
9	8	spcegv 2818	. . 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 2733	. . . . . 6 ⊢ 𝑥 ∈ V
12		bj-om 13972	. . . . . 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 1593	. . 3 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ∃𝑥 𝑥 = ω)
16		isset 2736	. . 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 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  ωcom 4574  Ind wind 13961

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdor 13851  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575  df-bdc 13876  df-bj-ind 13962

This theorem is referenced by:  bj-omex  13977