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Theorem ssel 2933
 Description: Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ssel (AB → (𝐶 A𝐶 B))

Proof of Theorem ssel
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 dfss2 2928 . . . . . 6 (ABx(x Ax B))
21biimpi 113 . . . . 5 (ABx(x Ax B))
3219.21bi 1447 . . . 4 (AB → (x Ax B))
43anim2d 320 . . 3 (AB → ((x = 𝐶 x A) → (x = 𝐶 x B)))
54eximdv 1757 . 2 (AB → (x(x = 𝐶 x A) → x(x = 𝐶 x B)))
6 df-clel 2033 . 2 (𝐶 Ax(x = 𝐶 x A))
7 df-clel 2033 . 2 (𝐶 Bx(x = 𝐶 x B))
85, 6, 73imtr4g 194 1 (AB → (𝐶 A𝐶 B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925 This theorem is referenced by:  ssel2  2934  sseli  2935  sseld  2938  sstr2  2946  ssralv  2998  ssrexv  2999  ralss  3000  rexss  3001  ssconb  3070  sscon  3071  ssdif  3072  unss1  3106  ssrin  3156  difin2  3193  reuss2  3211  reupick  3215  sssnm  3516  uniss  3592  ss2iun  3663  ssiun  3690  iinss  3699  disjss2  3739  disjss1  3742  pwnss  3903  sspwb  3943  ssopab2b  4004  soss  4042  sucssel  4127  ssorduni  4179  onnmin  4244  ssnel  4245  ssrel  4371  ssrel2  4373  ssrelrel  4383  xpss12  4388  cnvss  4451  dmss  4477  elreldm  4503  dmcosseq  4546  relssres  4591  iss  4597  resopab2  4598  issref  4650  ssrnres  4706  dfco2a  4764  cores  4767  funssres  4885  fununi  4910  funimaexglem  4925  dfimafn  5165  funimass4  5167  funimass3  5226  dff4im  5256  funfvima2  5334  funfvima3  5335  f1elima  5355  riotass2  5437  ssoprab2b  5504  resoprab2  5540  releldm2  5753  reldmtpos  5809  dmtpos  5812  rdgss  5910  eqreznegel  8305  negm  8306  iccsupr  8585  bdop  9310  bj-nnen2lp  9388
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