Theorem bj-omssind 13970

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 2312	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfv 1521	. . 3 ⊢ Ⅎ𝑥Ind 𝐴
3		bj-indeq 13964	. . . 4 ⊢ (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
4	3	biimprd 157	. . 3 ⊢ (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
5	1, 2, 4	bj-intabssel1 13825	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6		bj-dfom 13968	. . 3 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥}
7	6	sseq1i 3173	. 2 ⊢ (ω ⊆ 𝐴 ↔ ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)
8	5, 7	syl6ibr 161	1 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  {cab 2156   ⊆ wss 3121  ∩ cint 3831  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-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-ext 2152

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-v 2732  df-in 3127  df-ss 3134  df-int 3832  df-iom 4575  df-bj-ind 13962

This theorem is referenced by:  bj-om  13972  peano5set  13975