Theorem pssnel 4421

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 4303	. . 3 ⊢ ((𝐵 ∖ 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
3	1, 2	sylib 218	. 2 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴))
4		eldif 3909	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
5	4	exbii 1849	. 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 1780   ∈ wcel 2113   ≠ wne 2930   ∖ cdif 3896   ⊊ wpss 3900  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-v 3440  df-dif 3902  df-ss 3916  df-pss 3919  df-nul 4284

This theorem is referenced by:  pssnn  9091  php  9129  php3  9131  inf3lem2  9536  infpssr  10216  ssfin4  10218  genpnnp  10914  ltexprlem1  10945  reclem2pr  10957  mrieqv2d  17560  lbspss  21032  lsmcv  21094  lidlnz  21195  obslbs  21683  nmoid  24684  spansncvi  31676  fvineqsneq  37556  lsat0cv  39232  osumcllem11N  40165  pexmidlem8N  40176  isomenndlem  46716