Theorem sseq0 4404

Description: A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
sseq0	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)

Proof of Theorem sseq0

Step	Hyp	Ref	Expression
1		sseq2 4006	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ ∅))
2		ss0 4403	. . 3 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
3	1, 2	biimtrdi 252	. 2 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 → 𝐴 = ∅))
4	3	impcom 406	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ⊆ wss 3947  ∅c0 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-dif 3950  df-ss 3964  df-nul 4326

This theorem is referenced by:  ssn0  4405  ssdifin0  4490  disjxiun  5150  f1un  6863  fsetexb  8893  undomOLD  9098  infn0  9341  fieq0  9464  infdifsn  9700  cantnff  9717  tc00  9791  hashun3  14401  strleun  17159  dmdprdsplit2lem  20045  2idlval  21240  ocvval  21663  pjfval  21704  opsrle  22054  pf1rcl  22340  en2top  22979  nrmsep  23352  isnrm3  23354  regsep2  23371  xkohaus  23648  kqdisj  23727  regr1lem  23734  alexsublem  24039  reconnlem1  24833  metdstri  24858  iundisj2  25569  left0s  27916  right0s  27917  0clwlk0  30065  disjxpin  32508  iundisj2f  32510  iundisj2fi  32699  1arithufdlem4  33422  cvmsss2  35102  cldbnd  36038  cntotbnd  37497  nna4b4nsq  42314  mapfzcons1  42374  onfrALTlem2  44222  onfrALTlem2VD  44565  nnuzdisj  44970  ssdisjd  48193  ssdisjdr  48194  sepnsepolem2  48256  sepnsepo  48257