Theorem ssnpss 1573

Description: Partial trichotomy law for subclasses.

Assertion

Ref	Expression
ssnpss	⊢ (A ⊆ B → ¬ B ⊂ A)

Proof of Theorem ssnpss

Step	Hyp	Ref	Expression
1		sspss 1569	. 2 ⊢ (A ⊆ B ↔ (A ⊂ B ∨ A = B))
2		pssn2lp 1571	. . . 4 ⊢ ¬ (A ⊂ B ∧ B ⊂ A)
3		imnan 207	. . . 4 ⊢ ((A ⊂ B → ¬ B ⊂ A) ↔ ¬ (A ⊂ B ∧ B ⊂ A))
4	2, 3	mpbir 165	. . 3 ⊢ (A ⊂ B → ¬ B ⊂ A)
5		pssirr 1570	. . . 4 ⊢ ¬ A ⊂ A
6		psseq1 1559	. . . 4 ⊢ (A = B → (A ⊂ A ↔ B ⊂ A))
7	5, 6	mtbii 538	. . 3 ⊢ (A = B → ¬ B ⊂ A)
8	4, 7	jaoi 275	. 2 ⊢ ((A ⊂ B ∨ A = B) → ¬ B ⊂ A)
9	1, 8	sylbi 174	1 ⊢ (A ⊆ B → ¬ B ⊂ A)

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   = wceq 1091   ⊆ wss 1487   ⊂ wpss 1488

This theorem is referenced by:  suplem2pr 3956  atcvat 5771

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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