Theorem bj-2inf 16559

Description: Two formulations of the axiom of infinity (see ax-infvn 16562 and bj-omex 16563) . (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 2231	. . . 4 ⊢ ω = ω
2		bj-om 16558	. . . 4 ⊢ (ω ∈ V → (ω = ω ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
3	1, 2	mpbii 148	. . 3 ⊢ (ω ∈ V → (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
4		bj-indeq 16550	. . . . 5 ⊢ (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5		sseq1 3250	. . . . . . 7 ⊢ (𝑥 = ω → (𝑥 ⊆ 𝑦 ↔ ω ⊆ 𝑦))
6	5	imbi2d 230	. . . . . 6 ⊢ (𝑥 = ω → ((Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦)))
7	6	albidv 1872	. . . . 5 ⊢ (𝑥 = ω → (∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
8	4, 7	anbi12d 473	. . . 4 ⊢ (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
9	8	spcegv 2894	. . 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 2805	. . . . . 6 ⊢ 𝑥 ∈ V
12		bj-om 16558	. . . . . 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 133	. . . 4 ⊢ ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → 𝑥 = ω)
15	14	eximi 1648	. . 3 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ∃𝑥 𝑥 = ω)
16		isset 2809	. . 3 ⊢ (ω ∈ V ↔ ∃𝑥 𝑥 = ω)
17	15, 16	sylibr 134	. 2 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ω ∈ V)
18	10, 17	impbii 126	1 ⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802   ⊆ wss 3200  ωcom 4688  Ind wind 16547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4215  ax-pr 4299  ax-un 4530  ax-bd0 16434  ax-bdor 16437  ax-bdex 16440  ax-bdeq 16441  ax-bdel 16442  ax-bdsb 16443  ax-bdsep 16505

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-pr 3676  df-uni 3894  df-int 3929  df-suc 4468  df-iom 4689  df-bdc 16462  df-bj-ind 16548

This theorem is referenced by:  bj-omex  16563