Theorem bj-om 15429

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 15426	. . . 4 ⊢ Ind ω
2		bj-indeq 15421	. . . 4 ⊢ (𝐴 = ω → (Ind 𝐴 ↔ Ind ω))
3	1, 2	mpbiri 168	. . 3 ⊢ (𝐴 = ω → Ind 𝐴)
4		vex 2763	. . . . . 6 ⊢ 𝑥 ∈ V
5		bj-omssind 15427	. . . . . 6 ⊢ (𝑥 ∈ V → (Ind 𝑥 → ω ⊆ 𝑥))
6	4, 5	ax-mp 5	. . . . 5 ⊢ (Ind 𝑥 → ω ⊆ 𝑥)
7		sseq1 3202	. . . . 5 ⊢ (𝐴 = ω → (𝐴 ⊆ 𝑥 ↔ ω ⊆ 𝑥))
8	6, 7	imbitrrid 156	. . . 4 ⊢ (𝐴 = ω → (Ind 𝑥 → 𝐴 ⊆ 𝑥))
9	8	alrimiv 1885	. . 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 15428	. . . . . . 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 15427	. . . . 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 3194	. . 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 2164  Vcvv 2760   ⊆ wss 3153  ωcom 4622  Ind wind 15418

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4155  ax-pr 4238  ax-un 4464  ax-bd0 15305  ax-bdor 15308  ax-bdex 15311  ax-bdeq 15312  ax-bdel 15313  ax-bdsb 15314  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4402  df-iom 4623  df-bdc 15333  df-bj-ind 15419

This theorem is referenced by:  bj-2inf  15430  bj-inf2vn  15466  bj-inf2vn2  15467