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Theorem tfinds 6694
Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
tfinds.1
tfinds.2
tfinds.3
tfinds.4
tfinds.5
tfinds.6
tfinds.7
Assertion
Ref Expression
tfinds
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem tfinds
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 tfinds.2 . 2
2 tfinds.4 . 2
3 dflim3 6682 . . . . 5
43notbii 296 . . . 4
5 iman 424 . . . . 5
6 eloni 4893 . . . . . . 7
7 pm2.27 39 . . . . . . 7
86, 7syl 16 . . . . . 6
9 tfinds.5 . . . . . . . . 9
10 tfinds.1 . . . . . . . . 9
119, 10mpbiri 233 . . . . . . . 8
1211a1d 25 . . . . . . 7
13 nfra1 2838 . . . . . . . . 9
14 nfv 1707 . . . . . . . . 9
1513, 14nfim 1920 . . . . . . . 8
16 vex 3112 . . . . . . . . . . . . 13
1716sucid 4962 . . . . . . . . . . . 12
181rspcv 3206 . . . . . . . . . . . 12
1917, 18ax-mp 5 . . . . . . . . . . 11
20 tfinds.6 . . . . . . . . . . 11
2119, 20syl5 32 . . . . . . . . . 10
22 raleq 3054 . . . . . . . . . . . 12
23 nfv 1707 . . . . . . . . . . . . . . 15
2423, 1sbie 2149 . . . . . . . . . . . . . 14
25 sbequ 2117 . . . . . . . . . . . . . 14
2624, 25syl5bbr 259 . . . . . . . . . . . . 13
2726cbvralv 3084 . . . . . . . . . . . 12
28 cbvralsv 3095 . . . . . . . . . . . 12
2922, 27, 283bitr4g 288 . . . . . . . . . . 11
3029imbi1d 317 . . . . . . . . . 10
3121, 30syl5ibrcom 222 . . . . . . . . 9
32 tfinds.3 . . . . . . . . . . 11
3332biimprd 223 . . . . . . . . . 10
3433a1i 11 . . . . . . . . 9
3531, 34syldd 66 . . . . . . . 8
3615, 35rexlimi 2939 . . . . . . 7
3712, 36jaoi 379 . . . . . 6
388, 37syl6 33 . . . . 5
395, 38syl5bir 218 . . . 4
404, 39syl5bi 217 . . 3
41 tfinds.7 . . 3
4240, 41pm2.61d2 160 . 2
431, 2, 42tfis3 6692 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  [wsb 1739  e.wcel 1818  A.wral 2807  E.wrex 2808   c0 3784  Ordword 4882   con0 4883  Limwlim 4884  succsuc 4885
This theorem is referenced by:  tfindsg  6695  tfindes  6697  tfinds3  6699  oa0r  7207  om0r  7208  om1r  7211  oe1m  7213  oeoalem  7264  r1sdom  8213  r1tr  8215  alephon  8471  alephcard  8472  alephordi  8476  rdgprc  29227
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-sep 4573  ax-nul 4581  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-rab 2816  df-v 3111  df-sbc 3328  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-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889
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