Theorem fin23lem38 10266

Description: Lemma for fin23 10306. 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 7836	. . . . . . . 8 ⊢ (𝑑 ∈ ω → suc 𝑑 ∈ ω)
2		eqid 2737	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘suc 𝑑)
3		fveq2 6836	. . . . . . . . . . . . 13 ⊢ (𝑏 = suc 𝑑 → (𝑌‘𝑏) = (𝑌‘suc 𝑑))
4	3	rneqd 5889	. . . . . . . . . . . 12 ⊢ (𝑏 = suc 𝑑 → ran (𝑌‘𝑏) = ran (𝑌‘suc 𝑑))
5	4	unieqd 4864	. . . . . . . . . . 11 ⊢ (𝑏 = suc 𝑑 → ∪ ran (𝑌‘𝑏) = ∪ ran (𝑌‘suc 𝑑))
6	5	rspceeqv 3588	. . . . . . . . . 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 6849	. . . . . . . . . . . 12 ⊢ (𝑌‘suc 𝑑) ∈ V
9	8	rnex 7856	. . . . . . . . . . 11 ⊢ ran (𝑌‘suc 𝑑) ∈ V
10	9	uniex 7690	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑑) ∈ V
11		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏))
12	11	elrnmpt 5909	. . . . . . . . . 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 4906	. . . . . 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 10264	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∪ ran (𝑌‘suc 𝑑) ⊊ ∪ ran (𝑌‘𝑑))
25	18, 24	sspsstrd 4052	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑑))
26		dfpss2 4029	. . . . 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 3133	. 2 ⊢ (𝜑 → ¬ ∃𝑑 ∈ ω ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
30		fveq2 6836	. . . . . . 7 ⊢ (𝑏 = 𝑑 → (𝑌‘𝑏) = (𝑌‘𝑑))
31	30	rneqd 5889	. . . . . 6 ⊢ (𝑏 = 𝑑 → ran (𝑌‘𝑏) = ran (𝑌‘𝑑))
32	31	unieqd 4864	. . . . 5 ⊢ (𝑏 = 𝑑 → ∪ ran (𝑌‘𝑏) = ∪ ran (𝑌‘𝑑))
33	32	cbvmptv 5190	. . . 4 ⊢ (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = (𝑑 ∈ ω ↦ ∪ ran (𝑌‘𝑑))
34	33	elrnmpt 5909	. . 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 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890   ⊊ wpss 3891  𝒫 cpw 4542  ∪ cuni 4851  ∩ cint 4890   ↦ cmpt 5167  ran crn 5627   ↾ cres 5628  suc csuc 6321  –1-1→wf1 6491  ‘cfv 6494  (class class class)co 7362  ωcom 7812  reccrdg 8343   ↑_m cmap 8768

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 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344

This theorem is referenced by:  fin23lem39  10267