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Theorem dominf 8356
 Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8346. See dominfac 8479 for a version proved from ax-ac 8370. The axiom of Regularity is used for this proof, via inf3lem6 7617, and its use is necessary: otherwise the set or (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.)
Hypothesis
Ref Expression
dominf.1
Assertion
Ref Expression
dominf

Proof of Theorem dominf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dominf.1 . 2
2 neeq1 2615 . . . 4
3 id 21 . . . . 5
4 unieq 4048 . . . . 5
53, 4sseq12d 3363 . . . 4
62, 5anbi12d 693 . . 3
7 breq2 4241 . . 3
86, 7imbi12d 313 . 2
9 eqid 2442 . . . 4
10 eqid 2442 . . . 4
119, 10, 1, 1inf3lem6 7617 . . 3
12 vex 2965 . . . . 5
1312pwex 4411 . . . 4
1413f1dom 7158 . . 3
15 pwfi 7431 . . . . . . 7
1615biimpi 188 . . . . . 6
17 isfinite 7636 . . . . . 6
18 isfinite 7636 . . . . . 6
1916, 17, 183imtr3i 258 . . . . 5
2019con3i 130 . . . 4
2113domtriom 8354 . . . 4
2212domtriom 8354 . . . 4
2320, 21, 223imtr4i 259 . . 3
2411, 14, 233syl 19 . 2
251, 8, 24vtocl 3012 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   wceq 1653   wcel 1727   wne 2605  crab 2715  cvv 2962   cin 3305   wss 3306  c0 3613  cpw 3823  cuni 4039   class class class wbr 4237   cmpt 4291  com 4874   cres 4909  wf1 5480  crdg 6696   cdom 7136   csdm 7137  cfn 7138 This theorem is referenced by:  axgroth3  8737 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 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-reg 7589  ax-inf2 7625  ax-cc 8346 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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-card 7857  df-cda 8079
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