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Theorem bnj60 29505
 Description: Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj60.1
bnj60.2
bnj60.3
bnj60.4
Assertion
Ref Expression
bnj60
Distinct variable groups:   ,,,   ,   ,,,   ,,,
Allowed substitution hints:   (,)   (,,)   (,,)   (,,)

Proof of Theorem bnj60
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj60.1 . . . . 5
2 bnj60.2 . . . . 5
3 bnj60.3 . . . . 5
41, 2, 3bnj1497 29503 . . . 4
5 eqid 2438 . . . . . . . 8
61, 2, 3, 5bnj1311 29467 . . . . . . 7
763expia 1156 . . . . . 6
87ralrimiv 2790 . . . . 5
98ralrimiva 2791 . . . 4
10 biid 229 . . . . 5
11 biid 229 . . . . 5
1210, 5, 11bnj1383 29277 . . . 4
134, 9, 12sylancr 646 . . 3
14 bnj60.4 . . . 4
1514funeqi 5477 . . 3
1613, 15sylibr 205 . 2
171, 2, 3, 14bnj1498 29504 . 2
1816, 17bnj1422 29283 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cab 2424  wral 2707  wrex 2708   cin 3321   wss 3322  cop 3819  cuni 4017   cdm 4881   cres 4883   wfun 5451   wfn 5452  cfv 5457   c-bnj14 29126   w-bnj15 29130 This theorem is referenced by:  bnj1501  29510  bnj1523  29514 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-reg 7563  ax-inf2 7599 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-1o 6727  df-bnj17 29125  df-bnj14 29127  df-bnj13 29129  df-bnj15 29131  df-bnj18 29133  df-bnj19 29135
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