Theorem fin23lem38 10271

Description: Lemma for fin23 10311. 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 7841	. . . . . . . 8 ⊢ (𝑑 ∈ ω → suc 𝑑 ∈ ω)
2		eqid 2736	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘suc 𝑑)
3		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑏 = suc 𝑑 → (𝑌‘𝑏) = (𝑌‘suc 𝑑))
4	3	rneqd 5893	. . . . . . . . . . . 12 ⊢ (𝑏 = suc 𝑑 → ran (𝑌‘𝑏) = ran (𝑌‘suc 𝑑))
5	4	unieqd 4863	. . . . . . . . . . 11 ⊢ (𝑏 = suc 𝑑 → ∪ ran (𝑌‘𝑏) = ∪ ran (𝑌‘suc 𝑑))
6	5	rspceeqv 3587	. . . . . . . . . 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 692	. . . . . . . . 9 ⊢ (suc 𝑑 ∈ ω → ∃𝑏 ∈ ω ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘𝑏))
8		fvex 6853	. . . . . . . . . . . 12 ⊢ (𝑌‘suc 𝑑) ∈ V
9	8	rnex 7861	. . . . . . . . . . 11 ⊢ ran (𝑌‘suc 𝑑) ∈ V
10	9	uniex 7695	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑑) ∈ V
11		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏))
12	11	elrnmpt 5913	. . . . . . . . . 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 4905	. . . . . 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 10269	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∪ ran (𝑌‘suc 𝑑) ⊊ ∪ ran (𝑌‘𝑑))
25	18, 24	sspsstrd 4051	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑑))
26		dfpss2 4028	. . . . 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 497	. . . 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 3132	. 2 ⊢ (𝜑 → ¬ ∃𝑑 ∈ ω ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
30		fveq2 6840	. . . . . . 7 ⊢ (𝑏 = 𝑑 → (𝑌‘𝑏) = (𝑌‘𝑑))
31	30	rneqd 5893	. . . . . 6 ⊢ (𝑏 = 𝑑 → ran (𝑌‘𝑏) = ran (𝑌‘𝑑))
32	31	unieqd 4863	. . . . 5 ⊢ (𝑏 = 𝑑 → ∪ ran (𝑌‘𝑏) = ∪ ran (𝑌‘𝑑))
33	32	cbvmptv 5189	. . . 4 ⊢ (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = (𝑑 ∈ ω ↦ ∪ ran (𝑌‘𝑑))
34	33	elrnmpt 5913	. . 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 1540   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889   ⊊ wpss 3890  𝒫 cpw 4541  ∪ cuni 4850  ∩ cint 4889   ↦ cmpt 5166  ran crn 5632   ↾ cres 5633  suc csuc 6325  –1-1→wf1 6495  ‘cfv 6498  (class class class)co 7367  ωcom 7817  reccrdg 8348   ↑_m cmap 8773

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-sep 5231  ax-nul 5241  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-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-int 4890  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-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349

This theorem is referenced by:  fin23lem39  10272