Theorem npomex 10919

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 10916 and nsmallnq 10900). (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 3450	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ V)
2		prnmax 10918	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
3	2	ralrimiva 3129	. . . . 5 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
4		prpssnq 10913	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
5	4	pssssd 4040	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q)
6		ltsonq 10892	. . . . . . . . . 10 ⊢ <Q Or Q
7		soss 5559	. . . . . . . . . 10 ⊢ (𝐴 ⊆ Q → ( <Q Or Q → <Q Or 𝐴))
8	5, 6, 7	mpisyl 21	. . . . . . . . 9 ⊢ (𝐴 ∈ P → <Q Or 𝐴)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → <Q Or 𝐴)
10		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
11		prn0 10912	. . . . . . . . 9 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → 𝐴 ≠ ∅)
13		fimax2g 9196	. . . . . . . 8 ⊢ (( <Q Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦)
14	9, 10, 12, 13	syl3anc 1374	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦)
15		ralnex 3063	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
16	15	rexbii 3084	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <Q 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
17		rexnal 3089	. . . . . . . 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 218	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐴 ∈ Fin) → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)
20	19	ex 412	. . . . 5 ⊢ (𝐴 ∈ P → (𝐴 ∈ Fin → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
21	3, 20	mt2d 136	. . . 4 ⊢ (𝐴 ∈ P → ¬ 𝐴 ∈ Fin)
22		nelne1 3029	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → V ≠ Fin)
23	1, 21, 22	syl2anc 585	. . 3 ⊢ (𝐴 ∈ P → V ≠ Fin)
24	23	necomd 2987	. 2 ⊢ (𝐴 ∈ P → Fin ≠ V)
25		fineqv 9177	. . 3 ⊢ (¬ ω ∈ V ↔ Fin = V)
26	25	necon1abii 2980	. 2 ⊢ (Fin ≠ V ↔ ω ∈ V)
27	24, 26	sylib 218	1 ⊢ (𝐴 ∈ P → ω ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889  ∅c0 4273   class class class wbr 5085   Or wor 5538  ωcom 7817  Fincfn 8893  Qcnq 10775   <_Q cltq 10781  Pcnp 10782

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-ni 10795  df-mi 10797  df-lti 10798  df-ltpq 10833  df-enq 10834  df-nq 10835  df-ltnq 10841  df-np 10904

This theorem is referenced by: (None)