Theorem ssnpss 2152

Description: Partial trichotomy law for subclasses.

Assertion

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

Proof of Theorem ssnpss

Step	Hyp	Ref	Expression
1		sspss 2148	. 2 ⊢ (A ⊆ B ↔ (A ⊂ B ⋁ A = B))
2		pssn2lp 2150	. . . 4 ⊢ ¬ (A ⊂ B ⋀ B ⊂ A)
3		imnan 242	. . . 4 ⊢ ((A ⊂ B → ¬ B ⊂ A) ↔ ¬ (A ⊂ B ⋀ B ⊂ A))
4	2, 3	mpbir 190	. . 3 ⊢ (A ⊂ B → ¬ B ⊂ A)
5		pssirr 2149	. . . 4 ⊢ ¬ A ⊂ A
6		psseq1 2138	. . . 4 ⊢ (A = B → (A ⊂ A ↔ B ⊂ A))
7	5, 6	mtbii 718	. . 3 ⊢ (A = B → ¬ B ⊂ A)
8	4, 7	jaoi 341	. 2 ⊢ ((A ⊂ B ⋁ A = B) → ¬ B ⊂ A)
9	1, 8	sylbi 199	1 ⊢ (A ⊆ B → ¬ B ⊂ A)

Syntax hints:  ¬ wn 2   → wi 3   ⋁ wo 222   ⋀ wa 223   = wceq 958   ⊆ wss 2050   ⊂ wpss 2051

This theorem is referenced by:  suplem2pr 5174  atcvat 10308

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-in 2054  df-ss 2056  df-pss 2058

Copyright terms: Public domain