Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-ssom GIF version

Theorem bj-ssom 13553
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 3824 . . 3 (𝐴 {𝑦 ∣ Ind 𝑦} ↔ ∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴𝑥)
2 df-ral 2440 . . 3 (∀𝑥 ∈ {𝑦 ∣ Ind 𝑦}𝐴𝑥 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥))
3 vex 2715 . . . . . 6 𝑥 ∈ V
4 bj-indeq 13546 . . . . . 6 (𝑦 = 𝑥 → (Ind 𝑦 ↔ Ind 𝑥))
53, 4elab 2856 . . . . 5 (𝑥 ∈ {𝑦 ∣ Ind 𝑦} ↔ Ind 𝑥)
65imbi1i 237 . . . 4 ((𝑥 ∈ {𝑦 ∣ Ind 𝑦} → 𝐴𝑥) ↔ (Ind 𝑥𝐴𝑥))
76albii 1450 . . 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 13550 . . . 4 ω = {𝑦 ∣ Ind 𝑦}
109eqcomi 2161 . . 3 {𝑦 ∣ Ind 𝑦} = ω
1110sseq2i 3155 . 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 1333  wcel 2128  {cab 2143  wral 2435  wss 3102   cint 3808  ωcom 4550  Ind wind 13543
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-int 3809  df-iom 4551  df-bj-ind 13544
This theorem is referenced by:  bj-om  13554
  Copyright terms: Public domain W3C validator