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Theorem npomex 8620
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 8617 and nsmallnq 8601). (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 2796 . . . 4  |-  ( A  e.  P.  ->  A  e.  _V )
2 prnmax 8619 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  E. y  e.  A  x  <Q  y )
32ralrimiva 2626 . . . . 5  |-  ( A  e.  P.  ->  A. x  e.  A  E. y  e.  A  x  <Q  y )
4 prpssnq 8614 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  A  C.  Q. )
54pssssd 3273 . . . . . . . . . 10  |-  ( A  e.  P.  ->  A  C_ 
Q. )
6 ltsonq 8593 . . . . . . . . . 10  |-  <Q  Or  Q.
7 soss 4332 . . . . . . . . . 10  |-  ( A 
C_  Q.  ->  (  <Q  Or  Q.  ->  <Q  Or  A
) )
85, 6, 7ee10 1366 . . . . . . . . 9  |-  ( A  e.  P.  ->  <Q  Or  A )
98adantr 451 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  <Q  Or  A )
10 simpr 447 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  A  e.  Fin )
11 prn0 8613 . . . . . . . . 9  |-  ( A  e.  P.  ->  A  =/=  (/) )
1211adantr 451 . . . . . . . 8  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  A  =/=  (/) )
13 fimax2g 7103 . . . . . . . 8  |-  ( ( 
<Q  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x  <Q  y )
149, 10, 12, 13syl3anc 1182 . . . . . . 7  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  E. x  e.  A  A. y  e.  A  -.  x  <Q  y )
15 ralnex 2553 . . . . . . . . 9  |-  ( A. y  e.  A  -.  x  <Q  y  <->  -.  E. y  e.  A  x  <Q  y )
1615rexbii 2568 . . . . . . . 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 2554 . . . . . . . 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 240 . . . . . . 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 188 . . . . . 6  |-  ( ( A  e.  P.  /\  A  e.  Fin )  ->  -.  A. x  e.  A  E. y  e.  A  x  <Q  y
)
2019ex 423 . . . . 5  |-  ( A  e.  P.  ->  ( A  e.  Fin  ->  -.  A. x  e.  A  E. y  e.  A  x  <Q  y ) )
213, 20mt2d 109 . . . 4  |-  ( A  e.  P.  ->  -.  A  e.  Fin )
22 nelne1 2535 . . . 4  |-  ( ( A  e.  _V  /\  -.  A  e.  Fin )  ->  _V  =/=  Fin )
231, 21, 22syl2anc 642 . . 3  |-  ( A  e.  P.  ->  _V  =/=  Fin )
2423necomd 2529 . 2  |-  ( A  e.  P.  ->  Fin  =/=  _V )
25 fineqv 7078 . . 3  |-  ( -. 
om  e.  _V  <->  Fin  =  _V )
2625necon1abii 2497 . 2  |-  ( Fin 
=/=  _V  <->  om  e.  _V )
2724, 26sylib 188 1  |-  ( A  e.  P.  ->  om  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   class class class wbr 4023    Or wor 4313   omcom 4656   Fincfn 6863   Q.cnq 8474    <Q cltq 8480   P.cnp 8481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-ni 8496  df-mi 8498  df-lti 8499  df-ltpq 8534  df-enq 8535  df-nq 8536  df-ltnq 8542  df-np 8605
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