Theorem pssn2lp 3708

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 3693	. . . 4 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
2	1	simprbi 480	. . 3 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊆ 𝐴)
3		pssss 3702	. . 3 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
4	2, 3	nsyl 135	. 2 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴)
5		imnan 438	. 2 ⊢ ((𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴) ↔ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴))
6	4, 5	mpbi 220	1 ⊢ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ⊆ wss 3574   ⊊ wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by:  psstr  3711  cvnsym  29149