Theorem isf32lem5 10314

Description: Lemma for isfin3-2 10324. 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 10311	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
5	4	ralrimiva 3154	. 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 4035	. . . . . . 7 ⊢ 𝑆 ⊆ ω
8		nnunifi 9235	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω)
9	7, 8	mpan 700	. . . . . 6 ⊢ (𝑆 ∈ Fin → ∪ 𝑆 ∈ ω)
10	9	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω)
11		elssuni 4897	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝑆 → 𝑏 ⊆ ∪ 𝑆)
12		nnon 7852	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
13		omsson 7850	. . . . . . . . . . . . . . 15 ⊢ ω ⊆ On
14	13, 10	sselid 3934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ On)
15		ontri1 6380	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ On ∧ ∪ 𝑆 ∈ On) → (𝑏 ⊆ ∪ 𝑆 ↔ ¬ ∪ 𝑆 ∈ 𝑏))
16	12, 14, 15	syl2anr 606	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (𝑏 ⊆ ∪ 𝑆 ↔ ¬ ∪ 𝑆 ∈ 𝑏))
17	11, 16	imbitrid 246	. . . . . . . . . . . 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 458	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → ¬ 𝑏 ∈ 𝑆)
20	6	eleq2i 2854	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝑆 ↔ 𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)})
21	19, 20	sylnib 330	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → ¬ 𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)})
22		suceq 6414	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
23	22	fveq2d 6871	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘suc 𝑦) = (𝐹‘suc 𝑏))
24		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
25	23, 24	psseq12d 4050	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → ((𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦) ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
26	25	elrab3 3651	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
27	26	ad2antrl 738	. . . . . . . . 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 326	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → ¬ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))
29	28	expr 460	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (∪ 𝑆 ∈ 𝑏 → ¬ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
30		imnan 403	. . . . . . 7 ⊢ ((∪ 𝑆 ∈ 𝑏 → ¬ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ¬ (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
31	29, 30	sylib 220	. . . . . 6 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → ¬ (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
32	31	nrexdv 3157	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ¬ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
33		eleq1 2850	. . . . . . . . 9 ⊢ (𝑎 = ∪ 𝑆 → (𝑎 ∈ 𝑏 ↔ ∪ 𝑆 ∈ 𝑏))
34	33	anbi1d 640	. . . . . . . 8 ⊢ (𝑎 = ∪ 𝑆 → ((𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
35	34	rexbidv 3186	. . . . . . 7 ⊢ (𝑎 = ∪ 𝑆 → (∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
36	35	notbid 320	. . . . . 6 ⊢ (𝑎 = ∪ 𝑆 → (¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ¬ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
37	36	rspcev 3581	. . . . 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 593	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∃𝑎 ∈ ω ¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
39		rexnal 3114	. . . 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 220	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ¬ ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
41	40	ex 416	. 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 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  {crab 3414   ⊆ wss 3904   ⊊ wpss 3905  𝒫 cpw 4555  ∪ cuni 4865  ∩ cint 4905  ran crn 5648  Oncon0 6346  suc csuc 6348  ⟶wf 6517  ‘cfv 6521  ωcom 7846  Fincfn 8927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-om 7847  df-en 8928  df-fin 8931

This theorem is referenced by:  isf32lem6  10315  isf32lem7  10316  isf32lem8  10317