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Theorem cp 7561
 Description: Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 7555 that collapses a proper class into a set of minimum rank. The wff can be thought of as . Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.)
Assertion
Ref Expression
cp
Distinct variable groups:   ,,   ,,,
Allowed substitution hints:   (,)

Proof of Theorem cp
StepHypRef Expression
1 vex 2791 . . 3
21cplem2 7560 . 2
3 abn0 3473 . . . . 5
4 elin 3358 . . . . . . . 8
5 abid 2271 . . . . . . . . 9
65anbi1i 676 . . . . . . . 8
7 ancom 437 . . . . . . . 8
84, 6, 73bitri 262 . . . . . . 7
98exbii 1569 . . . . . 6
10 nfab1 2421 . . . . . . . 8
11 nfcv 2419 . . . . . . . 8
1210, 11nfin 3375 . . . . . . 7
1312n0f 3463 . . . . . 6
14 df-rex 2549 . . . . . 6
159, 13, 143bitr4i 268 . . . . 5
163, 15imbi12i 316 . . . 4
1716ralbii 2567 . . 3
1817exbii 1569 . 2
192, 18mpbi 199 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wex 1528   wcel 1684  cab 2269   wne 2446  wral 2543  wrex 2544   cin 3151  c0 3455 This theorem is referenced by:  bnd  7562 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
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