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

Theorem npomex 10019
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 , 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 10016 and nsmallnq 10000). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
Assertion
Ref Expression
npomex (𝐴P → ω ∈ V)

Proof of Theorem npomex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3361 . . . 4 (𝐴P𝐴 ∈ V)
2 prnmax 10018 . . . . . 6 ((𝐴P𝑥𝐴) → ∃𝑦𝐴 𝑥 <Q 𝑦)
32ralrimiva 3114 . . . . 5 (𝐴P → ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
4 prpssnq 10013 . . . . . . . . . . 11 (𝐴P𝐴Q)
54pssssd 3852 . . . . . . . . . 10 (𝐴P𝐴Q)
6 ltsonq 9992 . . . . . . . . . 10 <Q Or Q
7 soss 5188 . . . . . . . . . 10 (𝐴Q → ( <Q Or Q → <Q Or 𝐴))
85, 6, 7mpisyl 21 . . . . . . . . 9 (𝐴P → <Q Or 𝐴)
98adantr 466 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → <Q Or 𝐴)
10 simpr 471 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → 𝐴 ∈ Fin)
11 prn0 10012 . . . . . . . . 9 (𝐴P𝐴 ≠ ∅)
1211adantr 466 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → 𝐴 ≠ ∅)
13 fimax2g 8361 . . . . . . . 8 (( <Q Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦)
149, 10, 12, 13syl3anc 1475 . . . . . . 7 ((𝐴P𝐴 ∈ Fin) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦)
15 ralnex 3140 . . . . . . . . 9 (∀𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦𝐴 𝑥 <Q 𝑦)
1615rexbii 3188 . . . . . . . 8 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥 <Q 𝑦)
17 rexnal 3142 . . . . . . . 8 (∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
1816, 17bitri 264 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
1914, 18sylib 208 . . . . . 6 ((𝐴P𝐴 ∈ Fin) → ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
2019ex 397 . . . . 5 (𝐴P → (𝐴 ∈ Fin → ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦))
213, 20mt2d 133 . . . 4 (𝐴P → ¬ 𝐴 ∈ Fin)
22 nelne1 3038 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin)
231, 21, 22syl2anc 565 . . 3 (𝐴P → V ≠ Fin)
2423necomd 2997 . 2 (𝐴P → Fin ≠ V)
25 fineqv 8330 . . 3 (¬ ω ∈ V ↔ Fin = V)
2625necon1abii 2990 . 2 (Fin ≠ V ↔ ω ∈ V)
2724, 26sylib 208 1 (𝐴P → ω ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wcel 2144  wne 2942  wral 3060  wrex 3061  Vcvv 3349  wss 3721  c0 4061   class class class wbr 4784   Or wor 5169  ωcom 7211  Fincfn 8108  Qcnq 9875   <Q cltq 9881  Pcnp 9882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-omul 7717  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-ni 9895  df-mi 9897  df-lti 9898  df-ltpq 9933  df-enq 9934  df-nq 9935  df-ltnq 9941  df-np 10004
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator