Theorem isf32lem5 9773

Description: Lemma for isfin3-2 9783. 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 9770	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
5	4	ralrimiva 3182	. 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 4057	. . . . . . 7 ⊢ 𝑆 ⊆ ω
8		nnunifi 8763	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω)
9	7, 8	mpan 688	. . . . . 6 ⊢ (𝑆 ∈ Fin → ∪ 𝑆 ∈ ω)
10	9	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω)
11		elssuni 4861	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝑆 → 𝑏 ⊆ ∪ 𝑆)
12		nnon 7580	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
13		omsson 7578	. . . . . . . . . . . . . . 15 ⊢ ω ⊆ On
14	13, 10	sseldi 3965	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ On)
15		ontri1 6220	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ On ∧ ∪ 𝑆 ∈ On) → (𝑏 ⊆ ∪ 𝑆 ↔ ¬ ∪ 𝑆 ∈ 𝑏))
16	12, 14, 15	syl2anr 598	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (𝑏 ⊆ ∪ 𝑆 ↔ ¬ ∪ 𝑆 ∈ 𝑏))
17	11, 16	syl5ib 246	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (𝑏 ∈ 𝑆 → ¬ ∪ 𝑆 ∈ 𝑏))
18	17	con2d 136	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (∪ 𝑆 ∈ 𝑏 → ¬ 𝑏 ∈ 𝑆))
19	18	impr 457	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → ¬ 𝑏 ∈ 𝑆)
20	6	eleq2i 2904	. . . . . . . . . 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 6251	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
23	22	fveq2d 6669	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘suc 𝑦) = (𝐹‘suc 𝑏))
24		fveq2 6665	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
25	23, 24	psseq12d 4071	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → ((𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦) ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
26	25	elrab3 3681	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
27	26	ad2antrl 726	. . . . . . . . 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 459	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (∪ 𝑆 ∈ 𝑏 → ¬ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
30		imnan 402	. . . . . . 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 3270	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ¬ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
33		eleq1 2900	. . . . . . . . 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 3297	. . . . . . 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 3623	. . . . 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 586	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∃𝑎 ∈ ω ¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
39		rexnal 3238	. . . 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 415	. 2 ⊢ (𝜑 → (𝑆 ∈ Fin → ¬ ∀𝑎 ∈ ω ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
42	5, 41	mt2d 138	1 ⊢ (𝜑 → ¬ 𝑆 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  {crab 3142   ⊆ wss 3936   ⊊ wpss 3937  𝒫 cpw 4539  ∪ cuni 4832  ∩ cint 4869  ran crn 5551  Oncon0 6186  suc csuc 6188  ⟶wf 6346  ‘cfv 6350  ωcom 7574  Fincfn 8503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-om 7575  df-1o 8096  df-er 8283  df-en 8504  df-fin 8507

This theorem is referenced by:  isf32lem6  9774  isf32lem7  9775  isf32lem8  9776