Theorem isf32lem5 10317

Description: Lemma for isfin3-2 10327. 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 10314	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
5	4	ralrimiva 3126	. 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 4048	. . . . . . 7 ⊢ 𝑆 ⊆ ω
8		nnunifi 9245	. . . . . . 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 4904	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝑆 → 𝑏 ⊆ ∪ 𝑆)
12		nnon 7851	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
13		omsson 7849	. . . . . . . . . . . . . . 15 ⊢ ω ⊆ On
14	13, 10	sselid 3947	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ On)
15		ontri1 6369	. . . . . . . . . . . . . 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 2821	. . . . . . . . . 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 6403	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
23	22	fveq2d 6865	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘suc 𝑦) = (𝐹‘suc 𝑏))
24		fveq2 6861	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
25	23, 24	psseq12d 4063	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → ((𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦) ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
26	25	elrab3 3663	. . . . . . . . . 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 3129	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ¬ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
33		eleq1 2817	. . . . . . . . 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 3158	. . . . . . 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 3591	. . . . 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 3083	. . . 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 2109  ∀wral 3045  ∃wrex 3054  {crab 3408   ⊆ wss 3917   ⊊ wpss 3918  𝒫 cpw 4566  ∪ cuni 4874  ∩ cint 4913  ran crn 5642  Oncon0 6335  suc csuc 6337  ⟶wf 6510  ‘cfv 6514  ωcom 7845  Fincfn 8921

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-om 7846  df-en 8922  df-fin 8925

This theorem is referenced by:  isf32lem6  10318  isf32lem7  10319  isf32lem8  10320