Theorem npomex 10987

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 ℝ, 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 10984 and nsmallnq 10968). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
npomex	⊢ (𝐴 ∈ P → ω ∈ V)

Proof of Theorem npomex

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3492	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ V)
2		prnmax 10986	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
3	2	ralrimiva 3146	. . . . 5 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
4		prpssnq 10981	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
5	4	pssssd 4096	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
6		ltsonq 10960	. . . . . . . . . 10 ⊢ <Q Or Q
7		soss 5607	. . . . . . . . . 10 ⊢ (𝐴 ⊆ Q → ( <Q Or Q → <Q Or 𝐴))
8	5, 6, 7	mpisyl 21	. . . . . . . . 9 ⊢ (𝐴 ∈ P → <Q Or 𝐴)
9	8	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → <Q Or 𝐴)
10		simpr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
11		prn0 10980	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)
12	11	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅)
13		fimax2g 9285	. . . . . . . 8 ⊢ (( <Q Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦)
14	9, 10, 12, 13	syl3anc 1371	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦)
15		ralnex 3072	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
16	15	rexbii 3094	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
17		rexnal 3100	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
18	16, 17	bitri 274	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
19	14, 18	sylib 217	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
20	19	ex 413	. . . . 5 ⊢ (𝐴 ∈ P → (𝐴 ∈ Fin → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
21	3, 20	mt2d 136	. . . 4 ⊢ (𝐴 ∈ P → ¬ 𝐴 ∈ Fin)
22		nelne1 3039	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin)
23	1, 21, 22	syl2anc 584	. . 3 ⊢ (𝐴 ∈ P → V ≠ Fin)
24	23	necomd 2996	. 2 ⊢ (𝐴 ∈ P → Fin ≠ V)
25		fineqv 9259	. . 3 ⊢ (¬ ω ∈ V ↔ Fin = V)
26	25	necon1abii 2989	. 2 ⊢ (Fin ≠ V ↔ ω ∈ V)
27	24, 26	sylib 217	1 ⊢ (𝐴 ∈ P → ω ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ⊆ wss 3947  ∅c0 4321   class class class wbr 5147   Or wor 5586  ωcom 7851  Fincfn 8935  Qcnq 10843   <_Q cltq 10849  Pcnp 10850

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-omul 8467  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-ni 10863  df-mi 10865  df-lti 10866  df-ltpq 10901  df-enq 10902  df-nq 10903  df-ltnq 10909  df-np 10972

This theorem is referenced by: (None)