Theorem bj-omssind 14772

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 2319	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfv 1528	. . 3 ⊢ Ⅎ𝑥Ind 𝐴
3		bj-indeq 14766	. . . 4 ⊢ (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
4	3	biimprd 158	. . 3 ⊢ (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
5	1, 2, 4	bj-intabssel1 14627	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6		bj-dfom 14770	. . 3 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥}
7	6	sseq1i 3183	. 2 ⊢ (ω ⊆ 𝐴 ↔ ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)
8	5, 7	imbitrrdi 162	1 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  {cab 2163   ⊆ wss 3131  ∩ cint 3846  ωcom 4591  Ind wind 14763

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-in 3137  df-ss 3144  df-int 3847  df-iom 4592  df-bj-ind 14764

This theorem is referenced by:  bj-om  14774  peano5set  14777