Theorem bj-omssind 16298

Description: ω is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-omssind	⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))

Proof of Theorem bj-omssind

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2372	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfv 1574	. . 3 ⊢ Ⅎ𝑥Ind 𝐴
3		bj-indeq 16292	. . . 4 ⊢ (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
4	3	biimprd 158	. . 3 ⊢ (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
5	1, 2, 4	bj-intabssel1 16154	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6		bj-dfom 16296	. . 3 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥}
7	6	sseq1i 3250	. 2 ⊢ (ω ⊆ 𝐴 ↔ ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)
8	5, 7	imbitrrdi 162	1 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  {cab 2215   ⊆ wss 3197  ∩ cint 3923  ωcom 4682  Ind wind 16289

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-int 3924  df-iom 4683  df-bj-ind 16290

This theorem is referenced by:  bj-om  16300  peano5set  16303