Theorem pssnel 4431

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 4324	. . 3 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)
2		n0 4307	. . 3 ⊢ ((𝐵 ∖ 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
3	1, 2	sylib 221	. 2 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
4		eldif 3916	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
5	4	exbii 1881	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
6	3, 5	sylib 221	1 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960   ∖ cdif 3903   ⊊ wpss 3907  ∅c0 4286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3909  df-ss 3923  df-pss 3926  df-nul 4287

This theorem is used by:  pssnn  9160  php  9198  php3  9200  inf3lem2  9605  infpssr  10307  ssfin4  10309  genpnnp  11007  ltexprlem1  11038  reclem2pr  11050  mrieqv2d  17719  lbspss  21255  lsmcv  21317  lidlnz  21428  obslbs  21932  nmoid  24952  spansncvi  32077  fvineqsneq  38117  lsat0cv  39867  osumcllem11N  40800  pexmidlem8N  40811  isomenndlem  47304