ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ercl2 GIF version

Theorem ercl2 6542
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1 (𝜑𝑅 Er 𝑋)
ersym.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
ercl2 (𝜑𝐵𝑋)

Proof of Theorem ercl2
StepHypRef Expression
1 ersym.1 . 2 (𝜑𝑅 Er 𝑋)
2 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
31, 2ersym 6541 . 2 (𝜑𝐵𝑅𝐴)
41, 3ercl 6540 1 (𝜑𝐵𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148   class class class wbr 4000   Er wer 6526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-cnv 4631  df-dm 4633  df-er 6529
This theorem is referenced by:  qliftfun  6611  nqnq0pi  7425
  Copyright terms: Public domain W3C validator