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Theorem npomex 10395
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 10392 and nsmallnq 10376). (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 3489 . . . 4 (𝐴P𝐴 ∈ V)
2 prnmax 10394 . . . . . 6 ((𝐴P𝑥𝐴) → ∃𝑦𝐴 𝑥 <Q 𝑦)
32ralrimiva 3170 . . . . 5 (𝐴P → ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
4 prpssnq 10389 . . . . . . . . . . 11 (𝐴P𝐴Q)
54pssssd 4050 . . . . . . . . . 10 (𝐴P𝐴Q)
6 ltsonq 10368 . . . . . . . . . 10 <Q Or Q
7 soss 5466 . . . . . . . . . 10 (𝐴Q → ( <Q Or Q → <Q Or 𝐴))
85, 6, 7mpisyl 21 . . . . . . . . 9 (𝐴P → <Q Or 𝐴)
98adantr 484 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → <Q Or 𝐴)
10 simpr 488 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → 𝐴 ∈ Fin)
11 prn0 10388 . . . . . . . . 9 (𝐴P𝐴 ≠ ∅)
1211adantr 484 . . . . . . . 8 ((𝐴P𝐴 ∈ Fin) → 𝐴 ≠ ∅)
13 fimax2g 8740 . . . . . . . 8 (( <Q Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦)
149, 10, 12, 13syl3anc 1368 . . . . . . 7 ((𝐴P𝐴 ∈ Fin) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦)
15 ralnex 3224 . . . . . . . . 9 (∀𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦𝐴 𝑥 <Q 𝑦)
1615rexbii 3235 . . . . . . . 8 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥 <Q 𝑦)
17 rexnal 3226 . . . . . . . 8 (∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
1816, 17bitri 278 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
1914, 18sylib 221 . . . . . 6 ((𝐴P𝐴 ∈ Fin) → ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦)
2019ex 416 . . . . 5 (𝐴P → (𝐴 ∈ Fin → ¬ ∀𝑥𝐴𝑦𝐴 𝑥 <Q 𝑦))
213, 20mt2d 138 . . . 4 (𝐴P → ¬ 𝐴 ∈ Fin)
22 nelne1 3103 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin)
231, 21, 22syl2anc 587 . . 3 (𝐴P → V ≠ Fin)
2423necomd 3062 . 2 (𝐴P → Fin ≠ V)
25 fineqv 8709 . . 3 (¬ ω ∈ V ↔ Fin = V)
2625necon1abii 3055 . 2 (Fin ≠ V ↔ ω ∈ V)
2724, 26sylib 221 1 (𝐴P → ω ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wcel 2115  wne 3007  wral 3126  wrex 3127  Vcvv 3471  wss 3910  c0 4266   class class class wbr 5039   Or wor 5446  ωcom 7555  Fincfn 8484  Qcnq 10251   <Q cltq 10257  Pcnp 10258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-omul 8082  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-ni 10271  df-mi 10273  df-lti 10274  df-ltpq 10309  df-enq 10310  df-nq 10311  df-ltnq 10317  df-np 10380
This theorem is referenced by: (None)
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