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

Theorem npomex 10939
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 10936 and nsmallnq 10920). (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 3466 . . . 4 (𝐴P𝐴 ∈ V)
2 prnmax 10938 . . . . . 6 ((𝐴P𝑥𝐴) → ∃𝑦𝐴 𝑥 <Q 𝑦)
32ralrimiva 3144 . . . . 5 (𝐴P → ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
4 prpssnq 10933 . . . . . . . . . . 11 (𝐴P𝐴Q)
54pssssd 4062 . . . . . . . . . 10 (𝐴P𝐴Q)
6 ltsonq 10912 . . . . . . . . . 10 <Q Or Q
7 soss 5570 . . . . . . . . . 10 (𝐴Q → ( <Q Or Q → <Q Or 𝐴))
85, 6, 7mpisyl 21 . . . . . . . . 9 (𝐴P → <Q Or 𝐴)
98adantr 482 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → <Q Or 𝐴)
10 simpr 486 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → 𝐴 ∈ Fin)
11 prn0 10932 . . . . . . . . 9 (𝐴P𝐴 ≠ ∅)
1211adantr 482 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → 𝐴 ≠ ∅)
13 fimax2g 9240 . . . . . . . 8 (( <Q Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦)
149, 10, 12, 13syl3anc 1372 . . . . . . 7 ((𝐴P𝐴 ∈ Fin) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦)
15 ralnex 3076 . . . . . . . . 9 (∀𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦𝐴 𝑥 <Q 𝑦)
1615rexbii 3098 . . . . . . . 8 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥 <Q 𝑦)
17 rexnal 3104 . . . . . . . 8 (∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
1816, 17bitri 275 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
1914, 18sylib 217 . . . . . 6 ((𝐴P𝐴 ∈ Fin) → ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
2019ex 414 . . . . 5 (𝐴P → (𝐴 ∈ Fin → ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦))
213, 20mt2d 136 . . . 4 (𝐴P → ¬ 𝐴 ∈ Fin)
22 nelne1 3042 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin)
231, 21, 22syl2anc 585 . . 3 (𝐴P → V ≠ Fin)
2423necomd 3000 . 2 (𝐴P → Fin ≠ V)
25 fineqv 9214 . . 3 (¬ ω ∈ V ↔ Fin = V)
2625necon1abii 2993 . 2 (Fin ≠ V ↔ ω ∈ V)
2724, 26sylib 217 1 (𝐴P → ω ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3448  wss 3915  c0 4287   class class class wbr 5110   Or wor 5549  ωcom 7807  Fincfn 8890  Qcnq 10795   <Q cltq 10801  Pcnp 10802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-ni 10815  df-mi 10817  df-lti 10818  df-ltpq 10853  df-enq 10854  df-nq 10855  df-ltnq 10861  df-np 10924
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator