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Theorem bj-ssom 13061
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 3757 . . 3 (𝐴 {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴𝑥)
2 df-ral 2398 . . 3 (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥))
3 vex 2663 . . . . . 6 𝑥 ∈ V
4 bj-indeq 13054 . . . . . 6 (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥))
53, 4elab 2802 . . . . 5 (𝑥 ∈ {𝑦 ∣ Ind 𝑦} ↔ Ind 𝑥)
65imbi1i 237 . . . 4 ((𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥) ↔ (Ind 𝑥𝐴𝑥))
76albii 1431 . . 3 (∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥) ↔ ∀𝑥(Ind 𝑥𝐴𝑥))
81, 2, 73bitrri 206 . 2 (∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 {𝑦 ∣ Ind 𝑦})
9 bj-dfom 13058 . . . 4 ω = {𝑦 ∣ Ind 𝑦}
109eqcomi 2121 . . 3 {𝑦 ∣ Ind 𝑦} = ω
1110sseq2i 3094 . 2 (𝐴 {𝑦 ∣ Ind 𝑦} ↔ 𝐴 ⊆ ω)
128, 11bitri 183 1 (∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314  wcel 1465  {cab 2103  wral 2393  wss 3041   cint 3741  ωcom 4474  Ind wind 13051
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-in 3047  df-ss 3054  df-int 3742  df-iom 4475  df-bj-ind 13052
This theorem is referenced by:  bj-om  13062
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