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Theorem axdc2 8331
 Description: An apparent strengthening of ax-dc 8328 (but derived from it) which shows that there is a denumerable sequence for any function that maps elements of a set to nonempty subsets of such that for all . The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.)
Hypothesis
Ref Expression
axdc2.1
Assertion
Ref Expression
axdc2
Distinct variable groups:   ,,   ,,

Proof of Theorem axdc2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axdc2.1 . 2
2 eleq1 2498 . . . . 5
32adantr 453 . . . 4
4 fveq2 5730 . . . . . 6
54eleq2d 2505 . . . . 5
6 eleq1 2498 . . . . 5
75, 6sylan9bb 682 . . . 4
83, 7anbi12d 693 . . 3
98cbvopabv 4279 . 2
10 fveq2 5730 . . 3
1110cbvmptv 4302 . 2
121, 9, 11axdc2lem 8330 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551   wcel 1726   wne 2601  wral 2707  cvv 2958   cdif 3319  c0 3630  cpw 3801  csn 3816  copab 4267   cmpt 4268   csuc 4585  com 4847  wf 5452  cfv 5456 This theorem is referenced by:  axdc3lem4  8335 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-dc 8328 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-rab 2716  df-v 2960  df-sbc 3164  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-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-1o 6726
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