Theorem fin23lem38 10363

Description: Lemma for fin23 10403. The contradictory chain has no minimum. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Hypotheses

Ref	Expression
fin23lem33.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.f	⊢ (𝜑 → ℎ:ω–1-1→V)
fin23lem.g	⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
fin23lem.h	⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
fin23lem.i	⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)

Assertion

Ref	Expression
fin23lem38	⊢ (𝜑 → ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))

Distinct variable groups:   𝑎,𝑏,𝑔,𝑖,𝑗,𝑥,ℎ,𝐺   𝐹,𝑎   𝜑,𝑎,𝑏,𝑗   𝑌,𝑎,𝑏,𝑗

Allowed substitution hints:   𝜑(𝑥,𝑔,ℎ,𝑖)   𝐹(𝑥,𝑔,ℎ,𝑖,𝑗,𝑏)   𝑌(𝑥,𝑔,ℎ,𝑖)

Proof of Theorem fin23lem38

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7886	. . . . . . . 8 ⊢ (𝑑 ∈ ω → suc 𝑑 ∈ ω)
2		eqid 2735	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘suc 𝑑)
3		fveq2 6876	. . . . . . . . . . . . 13 ⊢ (𝑏 = suc 𝑑 → (𝑌‘𝑏) = (𝑌‘suc 𝑑))
4	3	rneqd 5918	. . . . . . . . . . . 12 ⊢ (𝑏 = suc 𝑑 → ran (𝑌‘𝑏) = ran (𝑌‘suc 𝑑))
5	4	unieqd 4896	. . . . . . . . . . 11 ⊢ (𝑏 = suc 𝑑 → ∪ ran (𝑌‘𝑏) = ∪ ran (𝑌‘suc 𝑑))
6	5	rspceeqv 3624	. . . . . . . . . 10 ⊢ ((suc 𝑑 ∈ ω ∧ ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘suc 𝑑)) → ∃𝑏 ∈ ω ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘𝑏))
7	2, 6	mpan2 691	. . . . . . . . 9 ⊢ (suc 𝑑 ∈ ω → ∃𝑏 ∈ ω ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘𝑏))
8		fvex 6889	. . . . . . . . . . . 12 ⊢ (𝑌‘suc 𝑑) ∈ V
9	8	rnex 7906	. . . . . . . . . . 11 ⊢ ran (𝑌‘suc 𝑑) ∈ V
10	9	uniex 7735	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑑) ∈ V
11		eqid 2735	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏))
12	11	elrnmpt 5938	. . . . . . . . . 10 ⊢ (∪ ran (𝑌‘suc 𝑑) ∈ V → (∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ↔ ∃𝑏 ∈ ω ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘𝑏)))
13	10, 12	ax-mp 5	. . . . . . . . 9 ⊢ (∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ↔ ∃𝑏 ∈ ω ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘𝑏))
14	7, 13	sylibr 234	. . . . . . . 8 ⊢ (suc 𝑑 ∈ ω → ∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))
15	1, 14	syl 17	. . . . . . 7 ⊢ (𝑑 ∈ ω → ∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))
16	15	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))
17		intss1 4939	. . . . . 6 ⊢ (∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) → ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊆ ∪ ran (𝑌‘suc 𝑑))
18	16, 17	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊆ ∪ ran (𝑌‘suc 𝑑))
19		fin23lem33.f	. . . . . 6 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
20		fin23lem.f	. . . . . 6 ⊢ (𝜑 → ℎ:ω–1-1→V)
21		fin23lem.g	. . . . . 6 ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
22		fin23lem.h	. . . . . 6 ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
23		fin23lem.i	. . . . . 6 ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)
24	19, 20, 21, 22, 23	fin23lem35 10361	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∪ ran (𝑌‘suc 𝑑) ⊊ ∪ ran (𝑌‘𝑑))
25	18, 24	sspsstrd 4086	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑑))
26		dfpss2 4063	. . . . 5 ⊢ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑑) ↔ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊆ ∪ ran (𝑌‘𝑑) ∧ ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑)))
27	26	simprbi 496	. . . 4 ⊢ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑑) → ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
28	25, 27	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
29	28	nrexdv 3135	. 2 ⊢ (𝜑 → ¬ ∃𝑑 ∈ ω ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
30		fveq2 6876	. . . . . . 7 ⊢ (𝑏 = 𝑑 → (𝑌‘𝑏) = (𝑌‘𝑑))
31	30	rneqd 5918	. . . . . 6 ⊢ (𝑏 = 𝑑 → ran (𝑌‘𝑏) = ran (𝑌‘𝑑))
32	31	unieqd 4896	. . . . 5 ⊢ (𝑏 = 𝑑 → ∪ ran (𝑌‘𝑏) = ∪ ran (𝑌‘𝑑))
33	32	cbvmptv 5225	. . . 4 ⊢ (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = (𝑑 ∈ ω ↦ ∪ ran (𝑌‘𝑑))
34	33	elrnmpt 5938	. . 3 ⊢ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) → (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ↔ ∃𝑑 ∈ ω ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑)))
35	34	ibi 267	. 2 ⊢ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) → ∃𝑑 ∈ ω ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
36	29, 35	nsyl 140	1 ⊢ (𝜑 → ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ⊆ wss 3926   ⊊ wpss 3927  𝒫 cpw 4575  ∪ cuni 4883  ∩ cint 4922   ↦ cmpt 5201  ran crn 5655   ↾ cres 5656  suc csuc 6354  –1-1→wf1 6528  ‘cfv 6531  (class class class)co 7405  ωcom 7861  reccrdg 8423   ↑_m cmap 8840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424

This theorem is referenced by:  fin23lem39  10364