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Theorem npomex 9418
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 9415 and nsmallnq 9399). (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 3053 . . . 4  |-  ( A  e.  P.  ->  A  e.  _V )
2 prnmax 9417 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. y  e.  A  x  <Q  y )
32ralrimiva 2801 . . . . 5  |-  ( A  e.  P.  ->  A. x  e.  A  E. y  e.  A  x  <Q  y )
4 prpssnq 9412 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  A  C. 
Q. )
54pssssd 3529 . . . . . . . . . 10  |-  ( A  e.  P.  ->  A  C_ 
Q. )
6 ltsonq 9391 . . . . . . . . . 10  |-  <Q  Or  Q.
7 soss 4772 . . . . . . . . . 10  |-  ( A 
C_  Q.  ->  (  <Q  Or  Q.  ->  <Q  Or  A
) )
85, 6, 7mpisyl 21 . . . . . . . . 9  |-  ( A  e.  P.  ->  <Q  Or  A )
98adantr 467 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  <Q  Or  A )
10 simpr 463 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  A  e.  Fin )
11 prn0 9411 . . . . . . . . 9  |-  ( A  e.  P.  ->  A  =/=  (/) )
1211adantr 467 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  A  =/=  (/) )
13 fimax2g 7814 . . . . . . . 8  |-  ( ( 
<Q  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x  <Q  y )
149, 10, 12, 13syl3anc 1267 . . . . . . 7  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  E. x  e.  A  A. y  e.  A  -.  x  <Q  y )
15 ralnex 2833 . . . . . . . . 9  |-  ( A. y  e.  A  -.  x  <Q  y  <->  -.  E. y  e.  A  x  <Q  y )
1615rexbii 2888 . . . . . . . 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 2835 . . . . . . . 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 253 . . . . . . 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 200 . . . . . 6  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  -.  A. x  e.  A  E. y  e.  A  x  <Q  y
)
2019ex 436 . . . . 5  |-  ( A  e.  P.  ->  ( A  e.  Fin  ->  -.  A. x  e.  A  E. y  e.  A  x  <Q  y ) )
213, 20mt2d 121 . . . 4  |-  ( A  e.  P.  ->  -.  A  e.  Fin )
22 nelne1 2719 . . . 4  |-  ( ( A  e.  _V  /\  -.  A  e.  Fin )  ->  _V  =/=  Fin )
231, 21, 22syl2anc 666 . . 3  |-  ( A  e.  P.  ->  _V  =/=  Fin )
2423necomd 2678 . 2  |-  ( A  e.  P.  ->  Fin  =/=  _V )
25 fineqv 7784 . . 3  |-  ( -. 
om  e.  _V  <->  Fin  =  _V )
2625necon1abii 2671 . 2  |-  ( Fin 
=/=  _V  <->  om  e.  _V )
2724, 26sylib 200 1  |-  ( A  e.  P.  ->  om  e.  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   _Vcvv 3044    C_ wss 3403   (/)c0 3730   class class class wbr 4401    Or wor 4753   omcom 6689   Fincfn 7566   Q.cnq 9274    <Q cltq 9280   P.cnp 9281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-omul 7184  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-ni 9294  df-mi 9296  df-lti 9297  df-ltpq 9332  df-enq 9333  df-nq 9334  df-ltnq 9340  df-np 9403
This theorem is referenced by: (None)
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