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

Theorem npomex 9395
 Description: A simplifying observation, and an indication of why any attempt to develop a theory of the real numbers without the Axiom of Infinity is doomed to failure: since every member of is an infinite set, the negation of Infinity implies that , and hence , is empty. (Note that this proof, which used the fact that Dedekind cuts have no maximum, could just as well have used that they have no minimum, since they are downward-closed by prcdnq 9392 and nsmallnq 9376). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
Assertion
Ref Expression
npomex

Proof of Theorem npomex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3118 . . . 4
2 prnmax 9394 . . . . . 6
32ralrimiva 2871 . . . . 5
4 prpssnq 9389 . . . . . . . . . . 11
54pssssd 3600 . . . . . . . . . 10
6 ltsonq 9368 . . . . . . . . . 10
7 soss 4823 . . . . . . . . . 10
85, 6, 7mpisyl 18 . . . . . . . . 9
98adantr 465 . . . . . . . 8
10 simpr 461 . . . . . . . 8
11 prn0 9388 . . . . . . . . 9
1211adantr 465 . . . . . . . 8
13 fimax2g 7786 . . . . . . . 8
149, 10, 12, 13syl3anc 1228 . . . . . . 7
15 ralnex 2903 . . . . . . . . 9
1615rexbii 2959 . . . . . . . 8
17 rexnal 2905 . . . . . . . 8
1816, 17bitri 249 . . . . . . 7
1914, 18sylib 196 . . . . . 6
2019ex 434 . . . . 5
213, 20mt2d 117 . . . 4
22 nelne1 2786 . . . 4
231, 21, 22syl2anc 661 . . 3
2423necomd 2728 . 2
25 fineqv 7755 . . 3
2625necon1abii 2719 . 2
2724, 26sylib 196 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808   cvv 3109  C_wss 3475   c0 3784   class class class wbr 4452  Orwor 4804   com 6700   cfn 7536   cnq 9251   cltq 9257   cnp 9258 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 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-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-oadd 7153  df-omul 7154  df-er 7330  df-en 7537  df-dom 7538  df-sdom 7539  df-fin 7540  df-ni 9271  df-mi 9273  df-lti 9274  df-ltpq 9309  df-enq 9310  df-nq 9311  df-ltnq 9317  df-np 9380
 Copyright terms: Public domain W3C validator