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Theorem bj-ssom 13971
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 3847 . . 3 (𝐴 {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴𝑥)
2 df-ral 2453 . . 3 (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥))
3 vex 2733 . . . . . 6 𝑥 ∈ V
4 bj-indeq 13964 . . . . . 6 (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥))
53, 4elab 2874 . . . . 5 (𝑥 ∈ {𝑦 ∣ Ind 𝑦} ↔ Ind 𝑥)
65imbi1i 237 . . . 4 ((𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥) ↔ (Ind 𝑥𝐴𝑥))
76albii 1463 . . 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 13968 . . . 4 ω = {𝑦 ∣ Ind 𝑦}
109eqcomi 2174 . . 3 {𝑦 ∣ Ind 𝑦} = ω
1110sseq2i 3174 . 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 1346  wcel 2141  {cab 2156  wral 2448  wss 3121   cint 3831  ωcom 4574  Ind wind 13961
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-int 3832  df-iom 4575  df-bj-ind 13962
This theorem is referenced by:  bj-om  13972
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