Theorem pssn2lp 3371

Description: Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
pssn2lp	⊢ ¬ (A ⊊ B ∧ B ⊊ A)

Proof of Theorem pssn2lp

Step	Hyp	Ref	Expression
1		dfpss3 3356	. . . 4 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ B ⊆ A))
2	1	simprbi 450	. . 3 ⊢ (A ⊊ B → ¬ B ⊆ A)
3		pssss 3365	. . 3 ⊢ (B ⊊ A → B ⊆ A)
4	2, 3	nsyl 113	. 2 ⊢ (A ⊊ B → ¬ B ⊊ A)
5		imnan 411	. 2 ⊢ ((A ⊊ B → ¬ B ⊊ A) ↔ ¬ (A ⊊ B ∧ B ⊊ A))
6	4, 5	mpbi 199	1 ⊢ ¬ (A ⊊ B ∧ B ⊊ A)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358   ⊆ wss 3258   ⊊ wpss 3259

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 proof 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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is used by:  psstr  3374