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Theorem fnwe2 35923
 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 6917 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 722 . . . . . 6
5 fnwe2.f . . . . . . 7
65adantr 467 . . . . . 6
7 fnwe2.r . . . . . . 7
87adantr 467 . . . . . 6
9 simprl 765 . . . . . 6
10 simprr 767 . . . . . 6
111, 2, 4, 6, 8, 9, 10fnwe2lem2 35921 . . . . 5
1211ex 436 . . . 4
1312alrimiv 1775 . . 3
14 df-fr 4796 . . 3
1513, 14sylibr 216 . 2
163adantlr 722 . . . 4
175adantr 467 . . . 4
187adantr 467 . . . 4
19 simprl 765 . . . 4
20 simprr 767 . . . 4
211, 2, 16, 17, 18, 19, 20fnwe2lem3 35922 . . 3
2221ralrimivva 2811 . 2
23 dfwe2 6613 . 2
2415, 22, 23sylanbrc 671 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 370   wa 371   w3o 985  wal 1444   wceq 1446   wcel 1889   wne 2624  wral 2739  wrex 2740  crab 2743   wss 3406  c0 3733   class class class wbr 4405  copab 4463   wfr 4793   wwe 4795   cres 4839  wf 5581  cfv 5585 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-un 6588 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593 This theorem is referenced by:  aomclem4  35927
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