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Theorem wsucex 28987
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 28973 . 2  |- wsuc ( R ,  A ,  X
)  =  sup ( Pred ( `' R ,  A ,  X ) ,  A ,  `' R
)
2 wsucex.1 . . . 4  |-  ( ph  ->  R  Or  A )
3 socnv 28799 . . . 4  |-  ( R  Or  A  ->  `' R  Or  A )
42, 3syl 16 . . 3  |-  ( ph  ->  `' R  Or  A
)
54supexd 7913 . 2  |-  ( ph  ->  sup ( Pred ( `' R ,  A ,  X ) ,  A ,  `' R )  e.  _V )
61, 5syl5eqel 2559 1  |-  ( ph  -> wsuc ( R ,  A ,  X )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   _Vcvv 3113    Or wor 4799   `'ccnv 4998   supcsup 7900   Predcpred 28848  wsuccwsuc 28971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rmo 2822  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-po 4800  df-so 4801  df-cnv 5007  df-sup 7901  df-wsuc 28973
This theorem is referenced by:  wsuclb  28989
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