Theorem 0pss 1730

Description: The null set is a proper subset of any non-empty set.

Assertion

Ref	Expression
0pss	⊢ (∅ ⊂ A ↔ ¬ A = ∅)

Proof of Theorem 0pss

Step	Hyp	Ref	Expression
1		dfpss2 1557	. . 3 ⊢ (∅ ⊂ A ↔ (∅ ⊆ A ∧ ¬ ∅ = A))
2		0ss 1725	. . 3 ⊢ ∅ ⊆ A
3	1, 2	mpbiran 547	. 2 ⊢ (∅ ⊂ A ↔ ¬ ∅ = A)
4		cleqcom 1103	. . 3 ⊢ (∅ = A ↔ A = ∅)
5	4	negbii 162	. 2 ⊢ (¬ ∅ = A ↔ ¬ A = ∅)
6	3, 5	bitr 151	1 ⊢ (∅ ⊂ A ↔ ¬ A = ∅)

Syntax hints:  ¬ wn 1   ↔ wb 127   = wceq 1091   ⊆ wss 1487   ⊂ wpss 1488  ∅c0 1707

This theorem is referenced by:  npss0 1731  php 3409  prn0 3887  genpn0 3900  1pr 3911  ltexprlem5 3940  reclem1pr 3950  suplem1pr 3955  infxpidmlem10 4942

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-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708

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