Theorem ss0b 1726

Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse.

Assertion

Ref	Expression
ss0b	⊢ (A ⊆ ∅ ↔ A = ∅)

Proof of Theorem ss0b

Step	Hyp	Ref	Expression
1		0ss 1725	. . 3 ⊢ ∅ ⊆ A
2		eqss 1516	. . . 4 ⊢ (A = ∅ ↔ (A ⊆ ∅ ∧ ∅ ⊆ A))
3	2	biimpr 134	. . 3 ⊢ ((A ⊆ ∅ ∧ ∅ ⊆ A) → A = ∅)
4	1, 3	mpan2 519	. 2 ⊢ (A ⊆ ∅ → A = ∅)
5		eqimss 1548	. 2 ⊢ (A = ∅ → A ⊆ ∅)
6	4, 5	impbi 139	1 ⊢ (A ⊆ ∅ ↔ A = ∅)

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ⊆ wss 1487  ∅c0 1707

This theorem is referenced by:  ss0 1727  un00 1728  undom 3342  kmlem5 3584  card0 3630  cf0 3705  infxpidmlem11 4943

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

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