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

Theorem npomex 9815
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 9812 and nsmallnq 9796). (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 3210 . . . 4 (𝐴P𝐴 ∈ V)
2 prnmax 9814 . . . . . 6 ((𝐴P𝑥𝐴) → ∃𝑦𝐴 𝑥 <Q 𝑦)
32ralrimiva 2965 . . . . 5 (𝐴P → ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
4 prpssnq 9809 . . . . . . . . . . 11 (𝐴P𝐴Q)
54pssssd 3702 . . . . . . . . . 10 (𝐴P𝐴Q)
6 ltsonq 9788 . . . . . . . . . 10 <Q Or Q
7 soss 5051 . . . . . . . . . 10 (𝐴Q → ( <Q Or Q → <Q Or 𝐴))
85, 6, 7mpisyl 21 . . . . . . . . 9 (𝐴P → <Q Or 𝐴)
98adantr 481 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → <Q Or 𝐴)
10 simpr 477 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → 𝐴 ∈ Fin)
11 prn0 9808 . . . . . . . . 9 (𝐴P𝐴 ≠ ∅)
1211adantr 481 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → 𝐴 ≠ ∅)
13 fimax2g 8203 . . . . . . . 8 (( <Q Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦)
149, 10, 12, 13syl3anc 1325 . . . . . . 7 ((𝐴P𝐴 ∈ Fin) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦)
15 ralnex 2991 . . . . . . . . 9 (∀𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦𝐴 𝑥 <Q 𝑦)
1615rexbii 3039 . . . . . . . 8 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥 <Q 𝑦)
17 rexnal 2994 . . . . . . . 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 450 . . . . 5 (𝐴P → (𝐴 ∈ Fin → ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦))
213, 20mt2d 131 . . . 4 (𝐴P → ¬ 𝐴 ∈ Fin)
22 nelne1 2889 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin)
231, 21, 22syl2anc 693 . . 3 (𝐴P → V ≠ Fin)
2423necomd 2848 . 2 (𝐴P → Fin ≠ V)
25 fineqv 8172 . . 3 (¬ ω ∈ V ↔ Fin = V)
2625necon1abii 2841 . 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 384  wcel 1989  wne 2793  wral 2911  wrex 2912  Vcvv 3198  wss 3572  c0 3913   class class class wbr 4651   Or wor 5032  ωcom 7062  Fincfn 7952  Qcnq 9671   <Q cltq 9677  Pcnp 9678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-omul 7562  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-ni 9691  df-mi 9693  df-lti 9694  df-ltpq 9729  df-enq 9730  df-nq 9731  df-ltnq 9737  df-np 9800
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator