Theorem pssnel 4476

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 4374	. . 3 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)
2		n0 4358	. . 3 ⊢ ((𝐵 ∖ 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
3	1, 2	sylib 218	. 2 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
4		eldif 3972	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
5	4	exbii 1844	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
6	3, 5	sylib 218	1 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937   ∖ cdif 3959   ⊊ wpss 3963  ∅c0 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-v 3479  df-dif 3965  df-ss 3979  df-pss 3982  df-nul 4339

This theorem is referenced by:  pssnn  9206  php  9244  php3  9246  phpOLD  9256  php3OLD  9258  inf3lem2  9666  infpssr  10345  ssfin4  10347  genpnnp  11042  ltexprlem1  11073  reclem2pr  11085  mrieqv2d  17683  lbspss  21098  lsmcv  21160  lidlnz  21269  obslbs  21767  nmoid  24778  spansncvi  31680  fvineqsneq  37394  lsat0cv  39014  osumcllem11N  39948  pexmidlem8N  39959  isomenndlem  46485