Theorem bj-omssind 16070

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 2350	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfv 1552	. . 3 ⊢ Ⅎ𝑥Ind 𝐴
3		bj-indeq 16064	. . . 4 ⊢ (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
4	3	biimprd 158	. . 3 ⊢ (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
5	1, 2, 4	bj-intabssel1 15926	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6		bj-dfom 16068	. . 3 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥}
7	6	sseq1i 3227	. 2 ⊢ (ω ⊆ 𝐴 ↔ ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)
8	5, 7	imbitrrdi 162	1 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2178  {cab 2193   ⊆ wss 3174  ∩ cint 3899  ωcom 4656  Ind wind 16061

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-in 3180  df-ss 3187  df-int 3900  df-iom 4657  df-bj-ind 16062

This theorem is referenced by:  bj-om  16072  peano5set  16075