Theorem wsuccl 31776

Description: If X is a set with an R successor in A, then its well-founded successor is a member of A. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)

Hypotheses

Ref	Expression
wsuccl.1	|- ( ph -> R We A )
wsuccl.2	|- ( ph -> R Se A )
wsuccl.3	|- ( ph -> X e. V )
wsuccl.4	|- ( ph -> E. y e. A X R y )

Assertion

Ref	Expression
wsuccl	|- ( ph -> wsuc ( R , A , X ) e. A )

Distinct variable groups:   y, R   y, A   y, X

Allowed substitution hints:   ph( y)   V( y)

Proof of Theorem wsuccl

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wsuc 31756	. 2 |- wsuc ( R , A , X ) = inf ( Pred ( `' R , A , X ) , A , R )
2		wsuccl.1	. . . 4 |- ( ph -> R We A )
3		weso 5105	. . . 4 |- ( R We A -> R Or A )
4	2, 3	syl 17	. . 3 |- ( ph -> R Or A )
5		wsuccl.2	. . . 4 |- ( ph -> R Se A )
6		wsuccl.3	. . . 4 |- ( ph -> X e. V )
7		wsuccl.4	. . . 4 |- ( ph -> E. y e. A X R y )
8	2, 5, 6, 7	wsuclem 31773	. . 3 |- ( ph -> E. a e. A ( A. b e. Pred ( `' R , A , X ) -. b R a /\ A. b e. A ( a R b -> E. c e. Pred ( `' R , A , X ) c R b ) ) )
9	4, 8	infcl 8394	. 2 |- ( ph -> inf ( Pred ( `' R , A , X ) , A , R ) e. A )
10	1, 9	syl5eqel 2705	1 |- ( ph -> wsuc ( R , A , X ) e. A )

Syntax hints:   -> wi 4   e. wcel 1990   E.wrex 2913   class class class wbr 4653   Or wor 5034   Se wse 5071   We wwe 5072   `'ccnv 5113   Predcpred 5679  infcinf 8347  wsuccwsuc 31752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-riota 6611  df-sup 8348  df-inf 8349  df-wsuc 31756

This theorem is referenced by: (None)