Theorem npomex 11005

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 11002 and nsmallnq 10986). (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 3471	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ V)
2		prnmax 11004	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
3	2	ralrimiva 3154	. . . . 5 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
4		prpssnq 10999	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
5	4	pssssd 4048	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
6		ltsonq 10978	. . . . . . . . . 10 ⊢ <Q Or Q
7		soss 5583	. . . . . . . . . 10 ⊢ (𝐴 ⊆ Q → ( <Q Or Q → <Q Or 𝐴))
8	5, 6, 7	mpisyl 22	. . . . . . . . 9 ⊢ (𝐴 ∈ P → <Q Or 𝐴)
9	8	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → <Q Or 𝐴)
10		simpr 490	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
11		prn0 10998	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)
12	11	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅)
13		fimax2g 9256	. . . . . . . 8 ⊢ (( <Q Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦)
14	9, 10, 12, 13	syl3anc 1398	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦)
15		ralnex 3088	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
16	15	rexbii 3109	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
17		rexnal 3114	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
18	16, 17	bitri 278	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
19	14, 18	sylib 221	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
20	19	ex 418	. . . . 5 ⊢ (𝐴 ∈ P → (𝐴 ∈ Fin → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
21	3, 20	mt2d 137	. . . 4 ⊢ (𝐴 ∈ P → ¬ 𝐴 ∈ Fin)
22		nelne1 3052	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin)
23	1, 21, 22	syl2anc 596	. . 3 ⊢ (𝐴 ∈ P → V ≠ Fin)
24	23	necomd 3010	. 2 ⊢ (𝐴 ∈ P → Fin ≠ V)
25		fineqv 9237	. . 3 ⊢ (¬ ω ∈ V ↔ Fin = V)
26	25	necon1abii 3003	. 2 ⊢ (Fin ≠ V ↔ ω ∈ V)
27	24, 26	sylib 221	1 ⊢ (𝐴 ∈ P → ω ∈ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103   Or wor 5562  ωcom 7862  Fincfn 8952  Qcnq 10861   <_Q cltq 10867  Pcnp 10868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-omul 8460  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-ni 10881  df-mi 10883  df-lti 10884  df-ltpq 10919  df-enq 10920  df-nq 10921  df-ltnq 10927  df-np 10990

This theorem is used by: (None)