Theorem npss0 1731

Description: No set is a proper subset of the empty set.

Assertion

Ref	Expression
npss0	⊢ ¬ A ⊂ ∅

Proof of Theorem npss0

Step	Hyp	Ref	Expression
1		cleqid 1102	. 2 ⊢ ∅ = ∅
2		pssss 1567	. . . . 5 ⊢ (A ⊂ ∅ → A ⊆ ∅)
3		ss0 1727	. . . . 5 ⊢ (A ⊆ ∅ → A = ∅)
4		psseq1 1559	. . . . 5 ⊢ (A = ∅ → (A ⊂ ∅ ↔ ∅ ⊂ ∅))
5	2, 3, 4	3syl 21	. . . 4 ⊢ (A ⊂ ∅ → (A ⊂ ∅ ↔ ∅ ⊂ ∅))
6	5	ibi 449	. . 3 ⊢ (A ⊂ ∅ → ∅ ⊂ ∅)
7		0pss 1730	. . 3 ⊢ (∅ ⊂ ∅ ↔ ¬ ∅ = ∅)
8	6, 7	sylib 173	. 2 ⊢ (A ⊂ ∅ → ¬ ∅ = ∅)
9	1, 8	mt2 96	1 ⊢ ¬ A ⊂ ∅

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

This theorem is referenced by:  pssnn 3428

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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