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Theorem wfrlem10 29569
 Description: Lemma for well-founded recursion. When is an minimal element of , then its predecessor class is equal to . (Contributed by Scott Fenton, 21-Apr-2011.)
Hypotheses
Ref Expression
wfrlem10.1
wfrlem10.2 wrecs
Assertion
Ref Expression
wfrlem10
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem wfrlem10
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 wfrlem10.2 . . . 4 wrecs
21wfrlem8 29567 . . 3
32biimpi 194 . 2
4 predss 29468 . . . 4
54a1i 11 . . 3
6 simpr 461 . . . . . 6
7 eldifn 3623 . . . . . . . . . 10
8 eleq1 2529 . . . . . . . . . . 11
98notbid 294 . . . . . . . . . 10
107, 9syl5ibrcom 222 . . . . . . . . 9
1110con2d 115 . . . . . . . 8
1211imp 429 . . . . . . 7
131wfrlem9 29568 . . . . . . . . . 10
1413adantl 466 . . . . . . . . 9
15 ssel 3493 . . . . . . . . . . . 12
1615con3d 133 . . . . . . . . . . 11
177, 16syl5com 30 . . . . . . . . . 10
1817adantr 465 . . . . . . . . 9
1914, 18mpd 15 . . . . . . . 8
20 eldifi 3622 . . . . . . . . 9
21 elpredg 29475 . . . . . . . . . 10
2221ancoms 453 . . . . . . . . 9
2320, 22sylan 471 . . . . . . . 8
2419, 23mtbid 300 . . . . . . 7
251wfrlem7 29566 . . . . . . . . 9
2625sseli 3495 . . . . . . . 8
27 wfrlem10.1 . . . . . . . . . 10
28 weso 4879 . . . . . . . . . 10
2927, 28ax-mp 5 . . . . . . . . 9
30 solin 4832 . . . . . . . . 9
3129, 30mpan 670 . . . . . . . 8
3226, 20, 31syl2anr 478 . . . . . . 7
3312, 24, 32ecase23d 1332 . . . . . 6
34 vex 3112 . . . . . . 7
35 vex 3112 . . . . . . . 8
3635elpred 29474 . . . . . . 7
3734, 36ax-mp 5 . . . . . 6
386, 33, 37sylanbrc 664 . . . . 5
3938ex 434 . . . 4
4039ssrdv 3505 . . 3
415, 40eqssd 3516 . 2
423, 41sylan9eqr 2520 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369   w3o 972   wceq 1395   wcel 1819  cvv 3109   cdif 3468   wss 3471  c0 3793   class class class wbr 4456   wor 4808   wwe 4846   cdm 5008  cpred 29460  wrecscwrecs 29552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-so 4810  df-we 4849  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-pred 29461  df-wrecs 29553 This theorem is referenced by:  wfrlem15  29574
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