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Theorem dfss4st 3313
Description: Subclass defined in terms of class difference. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
dfss4st (∀𝑥STAB 𝑥𝐴 → (𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem dfss4st
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2201 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21stbid 818 . . 3 (𝑥 = 𝑦 → (STAB 𝑥𝐴STAB 𝑦𝐴))
32cbvalv 1890 . 2 (∀𝑥STAB 𝑥𝐴 ↔ ∀𝑦STAB 𝑦𝐴)
4 nfa1 1522 . . . . 5 𝑦𝑦STAB 𝑦𝐴
5 nfcv 2282 . . . . 5 𝑦(𝐵 ∖ (𝐵𝐴))
6 nfcv 2282 . . . . 5 𝑦(𝐵𝐴)
7 eldif 3084 . . . . . . 7 (𝑦 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ (𝑦𝐵 ∧ ¬ 𝑦 ∈ (𝐵𝐴)))
8 elin 3263 . . . . . . . . . 10 (𝑦 ∈ (𝐵𝐴) ↔ (𝑦𝐵𝑦𝐴))
9 abai 550 . . . . . . . . . 10 ((𝑦𝐵𝑦𝐴) ↔ (𝑦𝐵 ∧ (𝑦𝐵𝑦𝐴)))
108, 9bitri 183 . . . . . . . . 9 (𝑦 ∈ (𝐵𝐴) ↔ (𝑦𝐵 ∧ (𝑦𝐵𝑦𝐴)))
11 imanst 874 . . . . . . . . . 10 (STAB 𝑦𝐴 → ((𝑦𝐵𝑦𝐴) ↔ ¬ (𝑦𝐵 ∧ ¬ 𝑦𝐴)))
1211anbi2d 460 . . . . . . . . 9 (STAB 𝑦𝐴 → ((𝑦𝐵 ∧ (𝑦𝐵𝑦𝐴)) ↔ (𝑦𝐵 ∧ ¬ (𝑦𝐵 ∧ ¬ 𝑦𝐴))))
1310, 12syl5bb 191 . . . . . . . 8 (STAB 𝑦𝐴 → (𝑦 ∈ (𝐵𝐴) ↔ (𝑦𝐵 ∧ ¬ (𝑦𝐵 ∧ ¬ 𝑦𝐴))))
14 eldif 3084 . . . . . . . . . 10 (𝑦 ∈ (𝐵𝐴) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐴))
1514notbii 658 . . . . . . . . 9 𝑦 ∈ (𝐵𝐴) ↔ ¬ (𝑦𝐵 ∧ ¬ 𝑦𝐴))
1615anbi2i 453 . . . . . . . 8 ((𝑦𝐵 ∧ ¬ 𝑦 ∈ (𝐵𝐴)) ↔ (𝑦𝐵 ∧ ¬ (𝑦𝐵 ∧ ¬ 𝑦𝐴)))
1713, 16syl6rbbr 198 . . . . . . 7 (STAB 𝑦𝐴 → ((𝑦𝐵 ∧ ¬ 𝑦 ∈ (𝐵𝐴)) ↔ 𝑦 ∈ (𝐵𝐴)))
187, 17syl5bb 191 . . . . . 6 (STAB 𝑦𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ 𝑦 ∈ (𝐵𝐴)))
1918sps 1518 . . . . 5 (∀𝑦STAB 𝑦𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ 𝑦 ∈ (𝐵𝐴)))
204, 5, 6, 19eqrd 3119 . . . 4 (∀𝑦STAB 𝑦𝐴 → (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴))
2120eqeq1d 2149 . . 3 (∀𝑦STAB 𝑦𝐴 → ((𝐵 ∖ (𝐵𝐴)) = 𝐴 ↔ (𝐵𝐴) = 𝐴))
22 sseqin2 3299 . . 3 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
2321, 22syl6rbbr 198 . 2 (∀𝑦STAB 𝑦𝐴 → (𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴))
243, 23sylbi 120 1 (∀𝑥STAB 𝑥𝐴 → (𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  STAB wstab 816  wal 1330   = wceq 1332  wcel 1481  cdif 3072  cin 3074  wss 3075
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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-stab 817  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077  df-in 3081  df-ss 3088
This theorem is referenced by:  sbthlemi3  6854
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