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Theorem dfpss4 3888
 Description: Alternate definition of proper subset. Theorem IX.4.21 of [Rosser] p. 236. (Contributed by SF, 19-Jan-2015.)
Assertion
Ref Expression
dfpss4 (AB ↔ (A B x B ¬ x A))
Distinct variable groups:   x,A   x,B

Proof of Theorem dfpss4
StepHypRef Expression
1 dfpss3 3355 . 2 (AB ↔ (A B ¬ B A))
2 dfss3 3263 . . . . 5 (B Ax B x A)
3 dfral2 2626 . . . . 5 (x B x A ↔ ¬ x B ¬ x A)
42, 3bitr2i 241 . . . 4 x B ¬ x AB A)
54con1bii 321 . . 3 B Ax B ¬ x A)
65anbi2i 675 . 2 ((A B ¬ B A) ↔ (A B x B ¬ x A))
71, 6bitri 240 1 (AB ↔ (A B x B ¬ x A))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261 This theorem is referenced by:  ssfin  4470
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