Theorem pssirr 4079

Description: Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Assertion

Ref	Expression
pssirr	⊢ ¬ 𝐴 ⊊ 𝐴

Proof of Theorem pssirr

Step	Hyp	Ref	Expression
1		pm3.24 405	. 2 ⊢ ¬ (𝐴 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐴)
2		dfpss3 4065	. 2 ⊢ (𝐴 ⊊ 𝐴 ↔ (𝐴 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐴))
3	1, 2	mtbir 325	1 ⊢ ¬ 𝐴 ⊊ 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 398   ⊆ wss 3938   ⊊ wpss 3939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-ne 3019  df-in 3945  df-ss 3954  df-pss 3956

This theorem is referenced by:  porpss  7455  ltsopr  10456