Theorem pssn2lp 4003

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 3988	. . . 4 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
2	1	simprbi 497	. . 3 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊆ 𝐴)
3		pssss 3997	. . 3 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
4	2, 3	nsyl 142	. 2 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴)
5		imnan 400	. 2 ⊢ ((𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴) ↔ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴))
6	4, 5	mpbi 231	1 ⊢ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ⊆ wss 3863   ⊊ wpss 3864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-ne 2985  df-in 3870  df-ss 3878  df-pss 3880

This theorem is referenced by:  psstr  4006  cvnsym  29763