Theorem pssn2lp 1571

Description: Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23.

Assertion

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

Proof of Theorem pssn2lp

Step	Hyp	Ref	Expression
1		pm3.24 496	. 2 ⊢ ¬ ((A ⊆ B ∧ B ⊆ A) ∧ ¬ (A ⊆ B ∧ B ⊆ A))
2		dfpss3 1558	. . . . 5 ⊢ (A ⊂ B ↔ (A ⊆ B ∧ ¬ B ⊆ A))
3		dfpss3 1558	. . . . 5 ⊢ (B ⊂ A ↔ (B ⊆ A ∧ ¬ A ⊆ B))
4	2, 3	anbi12i 369	. . . 4 ⊢ ((A ⊂ B ∧ B ⊂ A) ↔ ((A ⊆ B ∧ ¬ B ⊆ A) ∧ (B ⊆ A ∧ ¬ A ⊆ B)))
5		an42 389	. . . 4 ⊢ (((A ⊆ B ∧ ¬ B ⊆ A) ∧ (B ⊆ A ∧ ¬ A ⊆ B)) ↔ ((A ⊆ B ∧ B ⊆ A) ∧ (¬ A ⊆ B ∧ ¬ B ⊆ A)))
6	4, 5	bitr 151	. . 3 ⊢ ((A ⊂ B ∧ B ⊂ A) ↔ ((A ⊆ B ∧ B ⊆ A) ∧ (¬ A ⊆ B ∧ ¬ B ⊆ A)))
7		orc 225	. . . . . 6 ⊢ (¬ A ⊆ B → (¬ A ⊆ B ∨ ¬ B ⊆ A))
8	7	adantr 306	. . . . 5 ⊢ ((¬ A ⊆ B ∧ ¬ B ⊆ A) → (¬ A ⊆ B ∨ ¬ B ⊆ A))
9		ianor 253	. . . . 5 ⊢ (¬ (A ⊆ B ∧ B ⊆ A) ↔ (¬ A ⊆ B ∨ ¬ B ⊆ A))
10	8, 9	sylibr 175	. . . 4 ⊢ ((¬ A ⊆ B ∧ ¬ B ⊆ A) → ¬ (A ⊆ B ∧ B ⊆ A))
11	10	anim2i 270	. . 3 ⊢ (((A ⊆ B ∧ B ⊆ A) ∧ (¬ A ⊆ B ∧ ¬ B ⊆ A)) → ((A ⊆ B ∧ B ⊆ A) ∧ ¬ (A ⊆ B ∧ B ⊆ A)))
12	6, 11	sylbi 174	. 2 ⊢ ((A ⊂ B ∧ B ⊂ A) → ((A ⊆ B ∧ B ⊆ A) ∧ ¬ (A ⊆ B ∧ B ⊆ A)))
13	1, 12	mto 93	1 ⊢ ¬ (A ⊂ B ∧ B ⊂ A)

Syntax hints:  ¬ wn 1   ∨ wo 195   ∧ wa 196   ⊆ wss 1487   ⊂ wpss 1488

This theorem is referenced by:  ssnpss 1573  psstr 1574  cvnsymt 5722

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-in 1491  df-ss 1492  df-pss 1494

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