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Theorem wsucex 30296
Description: Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.)
Hypothesis
Ref Expression
wsucex.1  |-  ( ph  ->  R  Or  A )
Assertion
Ref Expression
wsucex  |-  ( ph  -> wsuc ( R ,  A ,  X )  e.  _V )

Proof of Theorem wsucex
StepHypRef Expression
1 df-wsuc 30282 . 2  |- wsuc ( R ,  A ,  X
)  =  sup ( Pred ( `' R ,  A ,  X ) ,  A ,  `' R
)
2 wsucex.1 . . . 4  |-  ( ph  ->  R  Or  A )
3 socnv 30192 . . . 4  |-  ( R  Or  A  ->  `' R  Or  A )
42, 3syl 17 . . 3  |-  ( ph  ->  `' R  Or  A
)
54supexd 7973 . 2  |-  ( ph  ->  sup ( Pred ( `' R ,  A ,  X ) ,  A ,  `' R )  e.  _V )
61, 5syl5eqel 2521 1  |-  ( ph  -> wsuc ( R ,  A ,  X )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   _Vcvv 3087    Or wor 4774   `'ccnv 4853   Predcpred 5398   supcsup 7960  wsuccwsuc 30280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rmo 2790  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-po 4775  df-so 4776  df-cnv 4862  df-sup 7962  df-wsuc 30282
This theorem is referenced by:  wsuclb  30298
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