Theorem pssnel 3616

Description: A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem pssnel

Step	Hyp	Ref	Expression
1		pssdif 3613	. . 3 ⊢ (A ⊊ B → (B ∖ A) ≠ ∅)
2		n0 3560	. . 3 ⊢ ((B ∖ A) ≠ ∅ ↔ ∃x x ∈ (B ∖ A))
3	1, 2	sylib 188	. 2 ⊢ (A ⊊ B → ∃x x ∈ (B ∖ A))
4		eldif 3222	. . 3 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
5	4	exbii 1582	. 2 ⊢ (∃x x ∈ (B ∖ A) ↔ ∃x(x ∈ B ∧ ¬ x ∈ A))
6	3, 5	sylib 188	1 ⊢ (A ⊊ B → ∃x(x ∈ B ∧ ¬ x ∈ A))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ≠ wne 2517   ∖ cdif 3207   ⊊ wpss 3259  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-pss 3262  df-nul 3552

This theorem is referenced by: (None)