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Theorem bj-ssom 11477
Description: A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ssom (∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-ssom
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssint 3699 . . 3 (𝐴 {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴𝑥)
2 df-ral 2364 . . 3 (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥))
3 vex 2622 . . . . . 6 𝑥 ∈ V
4 bj-indeq 11470 . . . . . 6 (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥))
53, 4elab 2758 . . . . 5 (𝑥 ∈ {𝑦 ∣ Ind 𝑦} ↔ Ind 𝑥)
65imbi1i 236 . . . 4 ((𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥) ↔ (Ind 𝑥𝐴𝑥))
76albii 1404 . . 3 (∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥) ↔ ∀𝑥(Ind 𝑥𝐴𝑥))
81, 2, 73bitrri 205 . 2 (∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 {𝑦 ∣ Ind 𝑦})
9 bj-dfom 11474 . . . 4 ω = {𝑦 ∣ Ind 𝑦}
109eqcomi 2092 . . 3 {𝑦 ∣ Ind 𝑦} = ω
1110sseq2i 3049 . 2 (𝐴 {𝑦 ∣ Ind 𝑦} ↔ 𝐴 ⊆ ω)
128, 11bitri 182 1 (∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wcel 1438  {cab 2074  wral 2359  wss 2997   cint 3683  ωcom 4395  Ind wind 11467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3003  df-ss 3010  df-int 3684  df-iom 4396  df-bj-ind 11468
This theorem is referenced by:  bj-om  11478
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