Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfri1dALT Unicode version

Theorem tfri1dALT 6295
 Description: Alternate proof of tfri1d 6279 in terms of tfr1on 6294. Although this does show that the tfr1on 6294 proof is general enough to also prove tfri1d 6279, the tfri1d 6279 proof is simpler in places because it does not need to deal with being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.)
Hypotheses
Ref Expression
tfri1dALT.1 recs
tfri1dALT.2
Assertion
Ref Expression
tfri1dALT
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem tfri1dALT
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrfun 6264 . . . 4 recs
2 tfri1dALT.1 . . . . 5 recs
32funeqi 5190 . . . 4 recs
41, 3mpbir 145 . . 3
54a1i 9 . 2
6 eqid 2157 . . . . . 6
76tfrlem8 6262 . . . . 5 recs
82dmeqi 4786 . . . . . 6 recs
9 ordeq 4332 . . . . . 6 recs recs
108, 9ax-mp 5 . . . . 5 recs
117, 10mpbir 145 . . . 4
12 ordsson 4450 . . . 4
1311, 12mp1i 10 . . 3
14 tfri1dALT.2 . . . . . . . . . 10
15 simpl 108 . . . . . . . . . . 11
1615alimi 1435 . . . . . . . . . 10
1714, 16syl 14 . . . . . . . . 9
181719.21bi 1538 . . . . . . . 8
1918adantr 274 . . . . . . 7
20 ordon 4444 . . . . . . . 8
2120a1i 9 . . . . . . 7
22 simpr 109 . . . . . . . . . . 11
2322alimi 1435 . . . . . . . . . 10
24 fveq2 5467 . . . . . . . . . . . 12
2524eleq1d 2226 . . . . . . . . . . 11
2625spv 1840 . . . . . . . . . 10
2714, 23, 263syl 17 . . . . . . . . 9
2827adantr 274 . . . . . . . 8
29283ad2ant1 1003 . . . . . . 7
30 suceloni 4459 . . . . . . . . 9
31 unon 4469 . . . . . . . . 9
3230, 31eleq2s 2252 . . . . . . . 8
3332adantl 275 . . . . . . 7
34 suceloni 4459 . . . . . . . 8
3534adantl 275 . . . . . . 7
362, 19, 21, 29, 33, 35tfr1on 6294 . . . . . 6
37 vex 2715 . . . . . . 7
3837sucid 4377 . . . . . 6
39 ssel2 3123 . . . . . 6
4036, 38, 39sylancl 410 . . . . 5
4140ex 114 . . . 4
4241ssrdv 3134 . . 3
4313, 42eqssd 3145 . 2
44 df-fn 5172 . 2
455, 43, 44sylanbrc 414 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104  wal 1333   wceq 1335   wcel 2128  cab 2143  wral 2435  wrex 2436  cvv 2712   wss 3102  cuni 3772   word 4322  con0 4323   csuc 4325   cdm 4585   cres 4587   wfun 5163   wfn 5164  cfv 5169  recscrecs 6248 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-recs 6249 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator