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Theorem npomex 9391
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  P. is an infinite set, the negation of Infinity implies that  P., and hence 
RR, 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 9388 and nsmallnq 9372). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
Assertion
Ref Expression
npomex  |-  ( A  e.  P.  ->  om  e.  _V )

Proof of Theorem npomex
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3118 . . . 4  |-  ( A  e.  P.  ->  A  e.  _V )
2 prnmax 9390 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. y  e.  A  x  <Q  y )
32ralrimiva 2871 . . . . 5  |-  ( A  e.  P.  ->  A. x  e.  A  E. y  e.  A  x  <Q  y )
4 prpssnq 9385 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  A  C. 
Q. )
54pssssd 3597 . . . . . . . . . 10  |-  ( A  e.  P.  ->  A  C_ 
Q. )
6 ltsonq 9364 . . . . . . . . . 10  |-  <Q  Or  Q.
7 soss 4827 . . . . . . . . . 10  |-  ( A 
C_  Q.  ->  (  <Q  Or  Q.  ->  <Q  Or  A
) )
85, 6, 7mpisyl 18 . . . . . . . . 9  |-  ( A  e.  P.  ->  <Q  Or  A )
98adantr 465 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  <Q  Or  A )
10 simpr 461 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  A  e.  Fin )
11 prn0 9384 . . . . . . . . 9  |-  ( A  e.  P.  ->  A  =/=  (/) )
1211adantr 465 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  A  =/=  (/) )
13 fimax2g 7784 . . . . . . . 8  |-  ( ( 
<Q  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x  <Q  y )
149, 10, 12, 13syl3anc 1228 . . . . . . 7  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  E. x  e.  A  A. y  e.  A  -.  x  <Q  y )
15 ralnex 2903 . . . . . . . . 9  |-  ( A. y  e.  A  -.  x  <Q  y  <->  -.  E. y  e.  A  x  <Q  y )
1615rexbii 2959 . . . . . . . 8  |-  ( E. x  e.  A  A. y  e.  A  -.  x  <Q  y  <->  E. x  e.  A  -.  E. y  e.  A  x  <Q  y )
17 rexnal 2905 . . . . . . . 8  |-  ( E. x  e.  A  -.  E. y  e.  A  x 
<Q  y  <->  -.  A. x  e.  A  E. y  e.  A  x  <Q  y )
1816, 17bitri 249 . . . . . . 7  |-  ( E. x  e.  A  A. y  e.  A  -.  x  <Q  y  <->  -.  A. x  e.  A  E. y  e.  A  x  <Q  y )
1914, 18sylib 196 . . . . . 6  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  -.  A. x  e.  A  E. y  e.  A  x  <Q  y
)
2019ex 434 . . . . 5  |-  ( A  e.  P.  ->  ( A  e.  Fin  ->  -.  A. x  e.  A  E. y  e.  A  x  <Q  y ) )
213, 20mt2d 117 . . . 4  |-  ( A  e.  P.  ->  -.  A  e.  Fin )
22 nelne1 2786 . . . 4  |-  ( ( A  e.  _V  /\  -.  A  e.  Fin )  ->  _V  =/=  Fin )
231, 21, 22syl2anc 661 . . 3  |-  ( A  e.  P.  ->  _V  =/=  Fin )
2423necomd 2728 . 2  |-  ( A  e.  P.  ->  Fin  =/=  _V )
25 fineqv 7754 . . 3  |-  ( -. 
om  e.  _V  <->  Fin  =  _V )
2625necon1abii 2719 . 2  |-  ( Fin 
=/=  _V  <->  om  e.  _V )
2724, 26sylib 196 1  |-  ( A  e.  P.  ->  om  e.  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109    C_ wss 3471   (/)c0 3793   class class class wbr 4456    Or wor 4808   omcom 6699   Fincfn 7535   Q.cnq 9247    <Q cltq 9253   P.cnp 9254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
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 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  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 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-omul 7153  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-ni 9267  df-mi 9269  df-lti 9270  df-ltpq 9305  df-enq 9306  df-nq 9307  df-ltnq 9313  df-np 9376
This theorem is referenced by: (None)
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