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Theorem fnwe2 27119
 Description: A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 6454 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
fnwe2.su
fnwe2.t
fnwe2.s
fnwe2.f
fnwe2.r
Assertion
Ref Expression
fnwe2
Distinct variable groups:   ,,   ,,   ,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   ()   ()   (,,)   ()

Proof of Theorem fnwe2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnwe2.su . . . . . 6
2 fnwe2.t . . . . . 6
3 fnwe2.s . . . . . . 7
43adantlr 696 . . . . . 6
5 fnwe2.f . . . . . . 7
65adantr 452 . . . . . 6
7 fnwe2.r . . . . . . 7
87adantr 452 . . . . . 6
9 simprl 733 . . . . . 6
10 simprr 734 . . . . . 6
111, 2, 4, 6, 8, 9, 10fnwe2lem2 27117 . . . . 5
1211ex 424 . . . 4
1312alrimiv 1641 . . 3
14 df-fr 4533 . . 3
1513, 14sylibr 204 . 2
163adantlr 696 . . . 4
175adantr 452 . . . 4
187adantr 452 . . . 4
19 simprl 733 . . . 4
20 simprr 734 . . . 4
211, 2, 16, 17, 18, 19, 20fnwe2lem3 27118 . . 3
2221ralrimivva 2790 . 2
23 dfwe2 4754 . 2
2415, 22, 23sylanbrc 646 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 358   wa 359   w3o 935  wal 1549   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698  crab 2701   wss 3312  c0 3620   class class class wbr 4204  copab 4257   wfr 4530   wwe 4532   cres 4872  wf 5442  cfv 5446 This theorem is referenced by:  aomclem4  27123 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-pr 4395  ax-un 4693 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-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454
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