Theorem pssnel 4471

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 4369	. . 3 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)
2		n0 4353	. . 3 ⊢ ((𝐵 ∖ 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
3	1, 2	sylib 218	. 2 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
4		eldif 3961	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
5	4	exbii 1848	. 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 1779   ∈ wcel 2108   ≠ wne 2940   ∖ cdif 3948   ⊊ wpss 3952  ∅c0 4333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-ss 3968  df-pss 3971  df-nul 4334

This theorem is referenced by:  pssnn  9208  php  9247  php3  9249  phpOLD  9259  php3OLD  9261  inf3lem2  9669  infpssr  10348  ssfin4  10350  genpnnp  11045  ltexprlem1  11076  reclem2pr  11088  mrieqv2d  17682  lbspss  21081  lsmcv  21143  lidlnz  21252  obslbs  21750  nmoid  24763  spansncvi  31671  fvineqsneq  37413  lsat0cv  39034  osumcllem11N  39968  pexmidlem8N  39979  isomenndlem  46545