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Theorem npomex 9375
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 9372 and nsmallnq 9356). (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 3122 . . . 4  |-  ( A  e.  P.  ->  A  e.  _V )
2 prnmax 9374 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. y  e.  A  x  <Q  y )
32ralrimiva 2878 . . . . 5  |-  ( A  e.  P.  ->  A. x  e.  A  E. y  e.  A  x  <Q  y )
4 prpssnq 9369 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  A  C. 
Q. )
54pssssd 3601 . . . . . . . . . 10  |-  ( A  e.  P.  ->  A  C_ 
Q. )
6 ltsonq 9348 . . . . . . . . . 10  |-  <Q  Or  Q.
7 soss 4818 . . . . . . . . . 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 9368 . . . . . . . . 9  |-  ( A  e.  P.  ->  A  =/=  (/) )
1211adantr 465 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  A  =/=  (/) )
13 fimax2g 7767 . . . . . . . 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 2910 . . . . . . . . 9  |-  ( A. y  e.  A  -.  x  <Q  y  <->  -.  E. y  e.  A  x  <Q  y )
1615rexbii 2965 . . . . . . . 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 2912 . . . . . . . 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 2796 . . . 4  |-  ( ( A  e.  _V  /\  -.  A  e.  Fin )  ->  _V  =/=  Fin )
231, 21, 22syl2anc 661 . . 3  |-  ( A  e.  P.  ->  _V  =/=  Fin )
2423necomd 2738 . 2  |-  ( A  e.  P.  ->  Fin  =/=  _V )
25 fineqv 7736 . . 3  |-  ( -. 
om  e.  _V  <->  Fin  =  _V )
2625necon1abii 2729 . 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 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447    Or wor 4799   omcom 6685   Fincfn 7517   Q.cnq 9231    <Q cltq 9237   P.cnp 9238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-omul 7136  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-ni 9251  df-mi 9253  df-lti 9254  df-ltpq 9289  df-enq 9290  df-nq 9291  df-ltnq 9297  df-np 9360
This theorem is referenced by: (None)
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