Theorem pssnel 1752

Description: A proper subclass has a member in one argument that's not in both.

Assertion

Ref	Expression
pssnel	⊢ (A ⊂ B → ∃x(x ∈ B ∧ ¬ x ∈ A))

Distinct variable group(s):   x,A   x,B

Proof of Theorem pssnel

Step	Hyp	Ref	Expression
1		dfpss2 1557	. . . 4 ⊢ (A ⊂ B ↔ (A ⊆ B ∧ ¬ A = B))
2		pssdifn0 1750	. . . 4 ⊢ ((A ⊆ B ∧ ¬ A = B) → ¬ (B ∖ A) = ∅)
3	1, 2	sylbi 174	. . 3 ⊢ (A ⊂ B → ¬ (B ∖ A) = ∅)
4		n0 1714	. . 3 ⊢ (¬ (B ∖ A) = ∅ ↔ ∃x x ∈ (B ∖ A))
5	3, 4	sylib 173	. 2 ⊢ (A ⊂ B → ∃x x ∈ (B ∖ A))
6		eldif 1496	. . 3 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
7	6	biex 733	. 2 ⊢ (∃x x ∈ (B ∖ A) ↔ ∃x(x ∈ B ∧ ¬ x ∈ A))
8	5, 7	sylib 173	1 ⊢ (A ⊂ B → ∃x(x ∈ B ∧ ¬ x ∈ A))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ⊆ wss 1487   ⊂ wpss 1488  ∅c0 1707

This theorem is referenced by:  php 3409  php3 3411  pssnn 3428  inf3lem2 3465  genpnnp 3902  ltexprlem1 3936  reclem1pr 3950  spansncv 5542

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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