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Theorem bnd 6446
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 4894), derived from the Collection Principle cp 6445. Its strength lies in the rather profound fact that does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom.
Assertion
Ref Expression
bnd
Distinct variable groups:   ,,   ,,,
Allowed substitution hints:   (,)

Proof of Theorem bnd
StepHypRef Expression
1 cp 6445 . . 3
2 ralim 2189 . . . 4
32eximi 1362 . . 3
41, 3ax-mp 8 . 2
5419.37aiv 1728 1
Colors of variables: wff set class
Syntax hints:   wi 4  wex 1335  wral 2127  wrex 2128
This theorem is referenced by:  bnd2 6447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1331  ax-6 1332  ax-7 1333  ax-gen 1334  ax-8 1418  ax-10 1419  ax-11 1420  ax-12 1421  ax-13 1422  ax-14 1423  ax-17 1430  ax-9 1445  ax-4 1451  ax-16 1629  ax-ext 1900  ax-rep 3481  ax-sep 3491  ax-nul 3500  ax-pow 3536  ax-pr 3560  ax-un 3836  ax-reg 6254  ax-inf2 6289
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3or 900  df-3an 901  df-ex 1336  df-sb 1591  df-eu 1818  df-mo 1819  df-clab 1906  df-cleq 1911  df-clel 1914  df-ne 2037  df-ral 2131  df-rex 2132  df-rab 2134  df-v 2329  df-sbc 2496  df-csb 2578  df-dif 2640  df-un 2642  df-in 2644  df-ss 2648  df-pss 2650  df-nul 2908  df-if 3015  df-pw 3075  df-sn 3092  df-pr 3093  df-tp 3094  df-op 3095  df-uni 3247  df-int 3281  df-iun 3320  df-iin 3321  df-br 3397  df-opab 3450  df-tr 3465  df-eprel 3646  df-id 3650  df-po 3655  df-so 3669  df-fr 3689  df-we 3705  df-ord 3721  df-on 3722  df-lim 3723  df-suc 3724  df-om 4001  df-xp 4048  df-rel 4049  df-cnv 4050  df-co 4051  df-dm 4052  df-rn 4053  df-res 4054  df-ima 4055  df-fun 4056  df-fn 4057  df-f 4058  df-fv 4062  df-mpt 5196  df-rdg 5535  df-r1 6332  df-rank 6333
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