Theorem npomex 10991

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 10988 and nsmallnq 10972). (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 3493	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ V)
2		prnmax 10990	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
3	2	ralrimiva 3147	. . . . 5 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
4		prpssnq 10985	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
5	4	pssssd 4098	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
6		ltsonq 10964	. . . . . . . . . 10 ⊢ <Q Or Q
7		soss 5609	. . . . . . . . . 10 ⊢ (𝐴 ⊆ Q → ( <Q Or Q → <Q Or 𝐴))
8	5, 6, 7	mpisyl 21	. . . . . . . . 9 ⊢ (𝐴 ∈ P → <Q Or 𝐴)
9	8	adantr 482	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → <Q Or 𝐴)
10		simpr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
11		prn0 10984	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)
12	11	adantr 482	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅)
13		fimax2g 9289	. . . . . . . 8 ⊢ (( <Q Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦)
14	9, 10, 12, 13	syl3anc 1372	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦)
15		ralnex 3073	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
16	15	rexbii 3095	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
17		rexnal 3101	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
18	16, 17	bitri 275	. . . . . . 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 414	. . . . 5 ⊢ (𝐴 ∈ P → (𝐴 ∈ Fin → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
21	3, 20	mt2d 136	. . . 4 ⊢ (𝐴 ∈ P → ¬ 𝐴 ∈ Fin)
22		nelne1 3040	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin)
23	1, 21, 22	syl2anc 585	. . 3 ⊢ (𝐴 ∈ P → V ≠ Fin)
24	23	necomd 2997	. 2 ⊢ (𝐴 ∈ P → Fin ≠ V)
25		fineqv 9263	. . 3 ⊢ (¬ ω ∈ V ↔ Fin = V)
26	25	necon1abii 2990	. 2 ⊢ (Fin ≠ V ↔ ω ∈ V)
27	24, 26	sylib 217	1 ⊢ (𝐴 ∈ P → ω ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ⊆ wss 3949  ∅c0 4323   class class class wbr 5149   Or wor 5588  ωcom 7855  Fincfn 8939  Qcnq 10847   <_Q cltq 10853  Pcnp 10854

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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-ni 10867  df-mi 10869  df-lti 10870  df-ltpq 10905  df-enq 10906  df-nq 10907  df-ltnq 10913  df-np 10976

This theorem is referenced by: (None)