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Theorem dfss4st 3279
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 2178 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21stbid 802 . . 3 (𝑥 = 𝑦 → (STAB 𝑥𝐴STAB 𝑦𝐴))
32cbvalv 1871 . 2 (∀𝑥STAB 𝑥𝐴 ↔ ∀𝑦STAB 𝑦𝐴)
4 nfa1 1506 . . . . 5 𝑦𝑦STAB 𝑦𝐴
5 nfcv 2258 . . . . 5 𝑦(𝐵 ∖ (𝐵𝐴))
6 nfcv 2258 . . . . 5 𝑦(𝐵𝐴)
7 eldif 3050 . . . . . . 7 (𝑦 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ (𝑦𝐵 ∧ ¬ 𝑦 ∈ (𝐵𝐴)))
8 elin 3229 . . . . . . . . . 10 (𝑦 ∈ (𝐵𝐴) ↔ (𝑦𝐵𝑦𝐴))
9 abai 534 . . . . . . . . . 10 ((𝑦𝐵𝑦𝐴) ↔ (𝑦𝐵 ∧ (𝑦𝐵𝑦𝐴)))
108, 9bitri 183 . . . . . . . . 9 (𝑦 ∈ (𝐵𝐴) ↔ (𝑦𝐵 ∧ (𝑦𝐵𝑦𝐴)))
11 imanst 858 . . . . . . . . . 10 (STAB 𝑦𝐴 → ((𝑦𝐵𝑦𝐴) ↔ ¬ (𝑦𝐵 ∧ ¬ 𝑦𝐴)))
1211anbi2d 459 . . . . . . . . 9 (STAB 𝑦𝐴 → ((𝑦𝐵 ∧ (𝑦𝐵𝑦𝐴)) ↔ (𝑦𝐵 ∧ ¬ (𝑦𝐵 ∧ ¬ 𝑦𝐴))))
1310, 12syl5bb 191 . . . . . . . 8 (STAB 𝑦𝐴 → (𝑦 ∈ (𝐵𝐴) ↔ (𝑦𝐵 ∧ ¬ (𝑦𝐵 ∧ ¬ 𝑦𝐴))))
14 eldif 3050 . . . . . . . . . 10 (𝑦 ∈ (𝐵𝐴) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐴))
1514notbii 642 . . . . . . . . 9 𝑦 ∈ (𝐵𝐴) ↔ ¬ (𝑦𝐵 ∧ ¬ 𝑦𝐴))
1615anbi2i 452 . . . . . . . 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 1502 . . . . 5 (∀𝑦STAB 𝑦𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ 𝑦 ∈ (𝐵𝐴)))
204, 5, 6, 19eqrd 3085 . . . 4 (∀𝑦STAB 𝑦𝐴 → (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴))
2120eqeq1d 2126 . . 3 (∀𝑦STAB 𝑦𝐴 → ((𝐵 ∖ (𝐵𝐴)) = 𝐴 ↔ (𝐵𝐴) = 𝐴))
22 sseqin2 3265 . . 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 800  wal 1314   = wceq 1316  wcel 1465  cdif 3038  cin 3040  wss 3041
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 588  ax-in2 589  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-stab 801  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054
This theorem is referenced by:  sbthlemi3  6815
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