Theorem bj-omssind 15581

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 2339	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfv 1542	. . 3 ⊢ Ⅎ𝑥Ind 𝐴
3		bj-indeq 15575	. . . 4 ⊢ (𝑥 = 𝐴 → (Ind 𝑥 ↔ Ind 𝐴))
4	3	biimprd 158	. . 3 ⊢ (𝑥 = 𝐴 → (Ind 𝐴 → Ind 𝑥))
5	1, 2, 4	bj-intabssel1 15436	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴))
6		bj-dfom 15579	. . 3 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥}
7	6	sseq1i 3209	. 2 ⊢ (ω ⊆ 𝐴 ↔ ∩ {𝑥 ∣ Ind 𝑥} ⊆ 𝐴)
8	5, 7	imbitrrdi 162	1 ⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  {cab 2182   ⊆ wss 3157  ∩ cint 3874  ωcom 4626  Ind wind 15572

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-int 3875  df-iom 4627  df-bj-ind 15573

This theorem is referenced by:  bj-om  15583  peano5set  15586