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

Theorem konigth 8071
 Description: Konig's Theorem. If for all , then , where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with regular unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting , this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.)
Hypotheses
Ref Expression
konigth.1
konigth.2
konigth.3
Assertion
Ref Expression
konigth
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem konigth
StepHypRef Expression
1 konigth.1 . 2
2 konigth.2 . 2
3 konigth.3 . 2
4 fveq2 5377 . . . . 5
54fveq1d 5379 . . . 4
65cbvmptv 4008 . . 3
76mpteq2i 4000 . 2
8 fveq2 5377 . . 3
98cbvmptv 4008 . 2
101, 2, 3, 7, 9konigthlem 8070 1
 Colors of variables: wff set class Syntax hints:   wi 6   wceq 1619   wcel 1621  wral 2509  cvv 2727  ciun 3803   class class class wbr 3920   cmpt 3974  cfv 4592  cixp 6703   csdm 6748 This theorem is referenced by:  pwcfsdom  8085 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-ac2 7973 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-er 6546  df-map 6660  df-ixp 6704  df-en 6750  df-dom 6751  df-sdom 6752  df-card 7456  df-acn 7459  df-ac 7627
 Copyright terms: Public domain W3C validator