Theorem nnullss 1880

Description: A non-empty class (even if proper) has a non-empty subset.

Assertion

Ref	Expression
nnullss	⊢ (¬ A = ∅ → ∃x(x ⊆ A ∧ ¬ x = ∅))

Distinct variable group(s):   x,A

Proof of Theorem nnullss

Step	Hyp	Ref	Expression
1		n0 1714	. 2 ⊢ (¬ A = ∅ ↔ ∃y y ∈ A)
2		visset 1350	. . . . 5 ⊢ y ∈ V
3	2	snss 1849	. . . 4 ⊢ (y ∈ A ↔ {y} ⊆ A)
4	2	snnz 1846	. . . . 5 ⊢ ¬ {y} = ∅
5		snex 1859	. . . . . 6 ⊢ {y} ∈ V
6		sseq1 1521	. . . . . . 7 ⊢ (x = {y} → (x ⊆ A ↔ {y} ⊆ A))
7		cleq1 1107	. . . . . . . 8 ⊢ (x = {y} → (x = ∅ ↔ {y} = ∅))
8	7	negbid 463	. . . . . . 7 ⊢ (x = {y} → (¬ x = ∅ ↔ ¬ {y} = ∅))
9	6, 8	anbi12d 476	. . . . . 6 ⊢ (x = {y} → ((x ⊆ A ∧ ¬ x = ∅) ↔ ({y} ⊆ A ∧ ¬ {y} = ∅)))
10	5, 9	cla4ev 1401	. . . . 5 ⊢ (({y} ⊆ A ∧ ¬ {y} = ∅) → ∃x(x ⊆ A ∧ ¬ x = ∅))
11	4, 10	mpan2 519	. . . 4 ⊢ ({y} ⊆ A → ∃x(x ⊆ A ∧ ¬ x = ∅))
12	3, 11	sylbi 174	. . 3 ⊢ (y ∈ A → ∃x(x ⊆ A ∧ ¬ x = ∅))
13	12	19.23aiv 952	. 2 ⊢ (∃y y ∈ A → ∃x(x ⊆ A ∧ ¬ x = ∅))
14	1, 13	sylbi 174	1 ⊢ (¬ A = ∅ → ∃x(x ⊆ A ∧ ¬ x = ∅))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ∅c0 1707  {csn 1808

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812

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