MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dominf Unicode version

Theorem dominf 8846
Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8836. See dominfac 8969 for a version proved from ax-ac 8860. The axiom of Regularity is used for this proof, via inf3lem6 8071, 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 2738 . . . 4
3 id 22 . . . . 5
4 unieq 4257 . . . . 5
53, 4sseq12d 3532 . . . 4
62, 5anbi12d 710 . . 3
7 breq2 4456 . . 3
86, 7imbi12d 320 . 2
9 eqid 2457 . . . 4
10 eqid 2457 . . . 4
119, 10, 1, 1inf3lem6 8071 . . 3
12 vex 3112 . . . . 5
1312pwex 4635 . . . 4
1413f1dom 7557 . . 3
15 pwfi 7835 . . . . . . 7
1615biimpi 194 . . . . . 6
17 isfinite 8090 . . . . . 6
18 isfinite 8090 . . . . . 6
1916, 17, 183imtr3i 265 . . . . 5
2019con3i 135 . . . 4
2113domtriom 8844 . . . 4
2212domtriom 8844 . . . 4
2320, 21, 223imtr4i 266 . . 3
2411, 14, 233syl 20 . 2
251, 8, 24vtocl 3161 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  {crab 2811   cvv 3109  i^icin 3474  C_wss 3475   c0 3784  ~Pcpw 4012  U.cuni 4249   class class class wbr 4452  e.cmpt 4510  |`cres 5006  -1-1->wf1 5590   com 6700  reccrdg 7094   cdom 7534   csdm 7535   cfn 7536
This theorem is referenced by:  axgroth3  9230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-reg 8039  ax-inf2 8079  ax-cc 8836
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-int 4287  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6701  df-1st 6800  df-2nd 6801  df-recs 7061  df-rdg 7095  df-1o 7149  df-2o 7150  df-oadd 7153  df-er 7330  df-map 7441  df-en 7537  df-dom 7538  df-sdom 7539  df-fin 7540  df-card 8341  df-cda 8569
  Copyright terms: Public domain W3C validator