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Theorem tz9.12 7705
 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 7702 through tz9.12lem3 7704. (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 2435 . . . 4
31, 2tz9.12lem2 7703 . . 3
43onsuci 4809 . 2
51, 2tz9.12lem3 7704 . 2
6 fveq2 5719 . . . 4
76eleq2d 2502 . . 3
87rspcev 3044 . 2
94, 5, 8sylancr 645 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  wral 2697  wrex 2698  crab 2701  cvv 2948  cuni 4007  cint 4042   cmpt 4258  con0 4573   csuc 4575  cima 4872  cfv 5445  cr1 7677 This theorem is referenced by:  tz9.13  7706  r1elss  7721 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-recs 6624  df-rdg 6659  df-r1 7679
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