Theorem nelne2 2879

Description: Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)

Assertion

Ref	Expression
nelne2	⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)

Proof of Theorem nelne2

Step	Hyp	Ref	Expression
1		eleq1 2676	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	biimpcd 238	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶))
3	2	necon3bd 2796	. 2 ⊢ (𝐴 ∈ 𝐶 → (¬ 𝐵 ∈ 𝐶 → 𝐴 ≠ 𝐵))
4	3	imp 444	1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-ne 2782

This theorem is referenced by:  nelelne  2880  elnelne2  2894  ac5num  8742  infpssrlem4  9011  fpwwe2lem13  9343  zgt1rpn0n1  11747  cats1un  13327  dprdfadd  18242  dprdcntz2  18260  lbsextlem4  18982  lindff1  19978  hauscmplem  21019  fileln0  21464  zcld  22424  dvcnvlem  23543  ppinprm  24678  chtnprm  24680  footex  25413  foot  25414  colperpexlem3  25424  mideulem2  25426  opphllem  25427  opphllem2  25440  lnopp2hpgb  25455  colhp  25462  lmieu  25476  trgcopy  25496  trgcopyeulem  25497  eupap1  26503  ordtconlem1  29298  esum2dlem  29481  subfacp1lem5  30420  heiborlem6  32785  llnle  33822  lplnle  33844  lhpexle1lem  34311  cdleme18b  34597  cdlemg46  35041  cdlemh  35123  bcc0  37561  fnchoice  38211  stoweidlem43  38936  zneoALTV  40118  elpwdifsn  40312