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Theorem tz9.12 8229
 Description: A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 8226 through tz9.12lem3 8228. (Contributed by NM, 22-Sep-2003.)
Hypothesis
Ref Expression
tz9.12.1
Assertion
Ref Expression
tz9.12
Distinct variable group:   ,,

Proof of Theorem tz9.12
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tz9.12.1 . . . 4
2 eqid 2457 . . . 4
31, 2tz9.12lem2 8227 . . 3
43onsuci 6673 . 2
51, 2tz9.12lem3 8228 . 2
6 fveq2 5871 . . . 4
76eleq2d 2527 . . 3
87rspcev 3210 . 2
94, 5, 8sylancr 663 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  {crab 2811   cvv 3109  U.cuni 4249  |^|cint 4286  e.cmpt 4510   con0 4883  succsuc 4885  "cima 5007  `cfv 5593   cr1 8201 This theorem is referenced by:  tz9.13  8230  r1elss  8245 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 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 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-int 4287  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-om 6701  df-recs 7061  df-rdg 7095  df-r1 8203
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