Theorem psstr 1574

Description: Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23.

Assertion

Ref	Expression
psstr	⊢ ((A ⊂ B ∧ B ⊂ C) → A ⊂ C)

Proof of Theorem psstr

Step	Hyp	Ref	Expression
1		pssss 1567	. . . 4 ⊢ (A ⊂ B → A ⊆ B)
2		pssss 1567	. . . 4 ⊢ (B ⊂ C → B ⊆ C)
3	1, 2	sylan9ss 1514	. . 3 ⊢ ((A ⊂ B ∧ B ⊂ C) → A ⊆ C)
4		pssn2lp 1571	. . . . 5 ⊢ ¬ (C ⊂ B ∧ B ⊂ C)
5		psseq1 1559	. . . . . 6 ⊢ (A = C → (A ⊂ B ↔ C ⊂ B))
6	5	anbi1d 469	. . . . 5 ⊢ (A = C → ((A ⊂ B ∧ B ⊂ C) ↔ (C ⊂ B ∧ B ⊂ C)))
7	4, 6	mtbiri 539	. . . 4 ⊢ (A = C → ¬ (A ⊂ B ∧ B ⊂ C))
8	7	con2i 89	. . 3 ⊢ ((A ⊂ B ∧ B ⊂ C) → ¬ A = C)
9	3, 8	jca 236	. 2 ⊢ ((A ⊂ B ∧ B ⊂ C) → (A ⊆ C ∧ ¬ A = C))
10		dfpss2 1557	. 2 ⊢ (A ⊂ C ↔ (A ⊆ C ∧ ¬ A = C))
11	9, 10	sylibr 175	1 ⊢ ((A ⊂ B ∧ B ⊂ C) → A ⊂ C)

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

This theorem is referenced by:  sspsstr 1575  psssstr 1576  inf3lem5 3468  zorn2 3612  ltsopr 3930

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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