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Theorem bj-ssom 9395
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 3622 . . 3 (A {y ∣ Ind y} ↔ x {y ∣ Ind y}Ax)
2 df-ral 2305 . . 3 (x {y ∣ Ind y}Axx(x {y ∣ Ind y} → Ax))
3 vex 2554 . . . . . 6 x V
4 bj-indeq 9388 . . . . . 6 (y = x → (Ind y ↔ Ind x))
53, 4elab 2681 . . . . 5 (x {y ∣ Ind y} ↔ Ind x)
65imbi1i 227 . . . 4 ((x {y ∣ Ind y} → Ax) ↔ (Ind xAx))
76albii 1356 . . 3 (x(x {y ∣ Ind y} → Ax) ↔ x(Ind xAx))
81, 2, 73bitrri 196 . 2 (x(Ind xAx) ↔ A {y ∣ Ind y})
9 bj-dfom 9392 . . . 4 𝜔 = {y ∣ Ind y}
109eqcomi 2041 . . 3 {y ∣ Ind y} = 𝜔
1110sseq2i 2964 . 2 (A {y ∣ Ind y} ↔ A ⊆ 𝜔)
128, 11bitri 173 1 (x(Ind xAx) ↔ A ⊆ 𝜔)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   wcel 1390  {cab 2023  wral 2300  wss 2911   cint 3606  𝜔com 4256  Ind wind 9385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-int 3607  df-iom 4257  df-bj-ind 9386
This theorem is referenced by:  bj-om  9396
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