Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  wessep Unicode version

Theorem wessep 4349
 Description: A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
Assertion
Ref Expression
wessep

Proof of Theorem wessep
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3003 . . . . . . 7
2 ssel 3003 . . . . . . 7
3 ssel 3003 . . . . . . 7
41, 2, 33anim123d 1251 . . . . . 6
54adantl 271 . . . . 5
65imdistani 434 . . . 4
7 wetrep 4144 . . . . . 6
87adantlr 461 . . . . 5
9 epel 4076 . . . . . 6
10 epel 4076 . . . . . 6
119, 10anbi12i 448 . . . . 5
12 epel 4076 . . . . 5
138, 11, 123imtr4g 203 . . . 4
146, 13syl 14 . . 3
1514ralrimivvva 2449 . 2
16 zfregfr 4345 . . 3
17 df-wetr 4118 . . 3
1816, 17mpbiran 882 . 2
1915, 18sylibr 132 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   w3a 920   wcel 1434  wral 2353   wss 2983   class class class wbr 3806   cep 4071   wfr 4112   wwe 4114 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-setind 4309 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-eprel 4073  df-frfor 4115  df-frind 4116  df-wetr 4118 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator