Theorem pssirr 1570

Description: Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23.

Assertion

Ref	Expression
pssirr	⊢ ¬ A ⊂ A

Proof of Theorem pssirr

Step	Hyp	Ref	Expression
1		pm3.24 496	. 2 ⊢ ¬ (A ⊆ A ∧ ¬ A ⊆ A)
2		dfpss3 1558	. 2 ⊢ (A ⊂ A ↔ (A ⊆ A ∧ ¬ A ⊆ A))
3	1, 2	mtbir 167	1 ⊢ ¬ A ⊂ A

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

This theorem is referenced by:  ssnpss 1573  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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