Theorem isf32lem5 10397

Description: Lemma for isfin3-2 10407. There are infinite decrease points. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Hypotheses

Ref	Expression
isf32lem.a	⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)
isf32lem.b	⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))
isf32lem.c	⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹)
isf32lem.d	⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)}

Assertion

Ref	Expression
isf32lem5	⊢ (𝜑 → ¬ 𝑆 ∈ Fin)

Distinct variable groups:   𝑥,𝑦,𝜑   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐺(𝑥,𝑦)

Proof of Theorem isf32lem5

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isf32lem.a	. . . 4 ⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)
2		isf32lem.b	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))
3		isf32lem.c	. . . 4 ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹)
4	1, 2, 3	isf32lem2 10394	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
5	4	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
6		isf32lem.d	. . . . . . . 8 ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)}
7	6	ssrab3 4082	. . . . . . 7 ⊢ 𝑆 ⊆ ω
8		nnunifi 9327	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω)
9	7, 8	mpan 690	. . . . . 6 ⊢ (𝑆 ∈ Fin → ∪ 𝑆 ∈ ω)
10	9	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω)
11		elssuni 4937	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝑆 → 𝑏 ⊆ ∪ 𝑆)
12		nnon 7893	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
13		omsson 7891	. . . . . . . . . . . . . . 15 ⊢ ω ⊆ On
14	13, 10	sselid 3981	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ On)
15		ontri1 6418	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ On ∧ ∪ 𝑆 ∈ On) → (𝑏 ⊆ ∪ 𝑆 ↔ ¬ ∪ 𝑆 ∈ 𝑏))
16	12, 14, 15	syl2anr 597	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (𝑏 ⊆ ∪ 𝑆 ↔ ¬ ∪ 𝑆 ∈ 𝑏))
17	11, 16	imbitrid 244	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (𝑏 ∈ 𝑆 → ¬ ∪ 𝑆 ∈ 𝑏))
18	17	con2d 134	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (∪ 𝑆 ∈ 𝑏 → ¬ 𝑏 ∈ 𝑆))
19	18	impr 454	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → ¬ 𝑏 ∈ 𝑆)
20	6	eleq2i 2833	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝑆 ↔ 𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)})
21	19, 20	sylnib 328	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → ¬ 𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)})
22		suceq 6450	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
23	22	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘suc 𝑦) = (𝐹‘suc 𝑏))
24		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
25	23, 24	psseq12d 4097	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → ((𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦) ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
26	25	elrab3 3693	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
27	26	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → (𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
28	21, 27	mtbid 324	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → ¬ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))
29	28	expr 456	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (∪ 𝑆 ∈ 𝑏 → ¬ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
30		imnan 399	. . . . . . 7 ⊢ ((∪ 𝑆 ∈ 𝑏 → ¬ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ¬ (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
31	29, 30	sylib 218	. . . . . 6 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → ¬ (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
32	31	nrexdv 3149	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ¬ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
33		eleq1 2829	. . . . . . . . 9 ⊢ (𝑎 = ∪ 𝑆 → (𝑎 ∈ 𝑏 ↔ ∪ 𝑆 ∈ 𝑏))
34	33	anbi1d 631	. . . . . . . 8 ⊢ (𝑎 = ∪ 𝑆 → ((𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
35	34	rexbidv 3179	. . . . . . 7 ⊢ (𝑎 = ∪ 𝑆 → (∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
36	35	notbid 318	. . . . . 6 ⊢ (𝑎 = ∪ 𝑆 → (¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ¬ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
37	36	rspcev 3622	. . . . 5 ⊢ ((∪ 𝑆 ∈ ω ∧ ¬ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))) → ∃𝑎 ∈ ω ¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
38	10, 32, 37	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∃𝑎 ∈ ω ¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
39		rexnal 3100	. . . 4 ⊢ (∃𝑎 ∈ ω ¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ¬ ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
40	38, 39	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ¬ ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
41	40	ex 412	. 2 ⊢ (𝜑 → (𝑆 ∈ Fin → ¬ ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
42	5, 41	mt2d 136	1 ⊢ (𝜑 → ¬ 𝑆 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436   ⊆ wss 3951   ⊊ wpss 3952  𝒫 cpw 4600  ∪ cuni 4907  ∩ cint 4946  ran crn 5686  Oncon0 6384  suc csuc 6386  ⟶wf 6557  ‘cfv 6561  ωcom 7887  Fincfn 8985

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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-en 8986  df-fin 8989

This theorem is referenced by:  isf32lem6  10398  isf32lem7  10399  isf32lem8  10400