Theorem bj-om 15667

Description: A set is equal to ω if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-om	⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥))))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-om

Step	Hyp	Ref	Expression
1		bj-omind 15664	. . . 4 ⊢ Ind ω
2		bj-indeq 15659	. . . 4 ⊢ (𝐴 = ω → (Ind 𝐴 ↔ Ind ω))
3	1, 2	mpbiri 168	. . 3 ⊢ (𝐴 = ω → Ind 𝐴)
4		vex 2766	. . . . . 6 ⊢ 𝑥 ∈ V
5		bj-omssind 15665	. . . . . 6 ⊢ (𝑥 ∈ V → (Ind 𝑥 → ω ⊆ 𝑥))
6	4, 5	ax-mp 5	. . . . 5 ⊢ (Ind 𝑥 → ω ⊆ 𝑥)
7		sseq1 3207	. . . . 5 ⊢ (𝐴 = ω → (𝐴 ⊆ 𝑥 ↔ ω ⊆ 𝑥))
8	6, 7	imbitrrid 156	. . . 4 ⊢ (𝐴 = ω → (Ind 𝑥 → 𝐴 ⊆ 𝑥))
9	8	alrimiv 1888	. . 3 ⊢ (𝐴 = ω → ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥))
10	3, 9	jca 306	. 2 ⊢ (𝐴 = ω → (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)))
11		bj-ssom 15666	. . . . . . 7 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω)
12	11	biimpi 120	. . . . . 6 ⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) → 𝐴 ⊆ ω)
13	12	adantl 277	. . . . 5 ⊢ ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → 𝐴 ⊆ ω)
14	13	a1i 9	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → 𝐴 ⊆ ω))
15		bj-omssind 15665	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
16	15	adantrd 279	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → ω ⊆ 𝐴))
17	14, 16	jcad 307	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → (𝐴 ⊆ ω ∧ ω ⊆ 𝐴)))
18		eqss 3199	. . 3 ⊢ (𝐴 = ω ↔ (𝐴 ⊆ ω ∧ ω ⊆ 𝐴))
19	17, 18	imbitrrdi 162	. 2 ⊢ (𝐴 ∈ 𝑉 → ((Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)) → 𝐴 = ω))
20	10, 19	impbid2 143	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  ωcom 4627  Ind wind 15656

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4160  ax-pr 4243  ax-un 4469  ax-bd0 15543  ax-bdor 15546  ax-bdex 15549  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552  ax-bdsep 15614

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-suc 4407  df-iom 4628  df-bdc 15571  df-bj-ind 15657

This theorem is referenced by:  bj-2inf  15668  bj-inf2vn  15704  bj-inf2vn2  15705