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Theorem ecexr 6010
Description: An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecexr (A [B]𝑅B V)

Proof of Theorem ecexr
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elimag 4587 . . . . 5 (A (𝑅 “ {B}) → (A (𝑅 “ {B}) ↔ x {B}x𝑅A))
21ibi 165 . . . 4 (A (𝑅 “ {B}) → x {B}x𝑅A)
3 df-ec 6007 . . . 4 [B]𝑅 = (𝑅 “ {B})
42, 3eleq2s 2105 . . 3 (A [B]𝑅x {B}x𝑅A)
5 df-rex 2281 . . . 4 (x {B}x𝑅Ax(x {B} x𝑅A))
6 simpl 102 . . . . . 6 ((x {B} x𝑅A) → x {B})
7 elsn 3354 . . . . . 6 (x {B} ↔ x = B)
86, 7sylib 127 . . . . 5 ((x {B} x𝑅A) → x = B)
98eximi 1464 . . . 4 (x(x {B} x𝑅A) → x x = B)
105, 9sylbi 114 . . 3 (x {B}x𝑅Ax x = B)
114, 10syl 14 . 2 (A [B]𝑅x x = B)
12 isset 2530 . 2 (B V ↔ x x = B)
1311, 12sylibr 137 1 (A [B]𝑅B V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1223  wex 1354   wcel 1366  wrex 2276  Vcvv 2526  {csn 3339   class class class wbr 3727  cima 4263  [cec 6003
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-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-xp 4266  df-cnv 4268  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-ec 6007
This theorem is referenced by:  relelec  6045  ecdmn0m  6047
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