Theorem pssnel 4426

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 4319	. . 3 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)
2		n0 4302	. . 3 ⊢ ((𝐵 ∖ 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
3	1, 2	sylib 221	. 2 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
4		eldif 4051	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
5	4	exbii 1880	. 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 1811   ∈ wcel 2146   ≠ wne 2958   ⊊ wpss 4015   ∖ cdif 4047  ∅c0 4280

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3455  df-ss 3900  df-pss 4016  df-dif 4049  df-nul 4281

This theorem is used by:  pssnn  9195  php  9233  php3  9235  inf3lem2  9640  infpssr  10396  ssfin4  10398  genpnnp  11100  ltexprlem1  11131  reclem2pr  11143  mrieqv2d  17825  lbspss  21369  lsmcv  21431  lidlnz  21542  obslbs  22048  nmoid  25073  spansncvi  32266  fvineqsneq  38475  lsat0cv  40230  osumcllem11N  41163  pexmidlem8N  41174  isomenndlem  47679