Theorem isf32lem5 9768

Description: Lemma for isfin3-2 9778. 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 9765	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
5	4	ralrimiva 3149	. 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 4008	. . . . . . 7 ⊢ 𝑆 ⊆ ω
8		nnunifi 8753	. . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω)
9	7, 8	mpan 689	. . . . . 6 ⊢ (𝑆 ∈ Fin → ∪ 𝑆 ∈ ω)
10	9	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω)
11		elssuni 4830	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝑆 → 𝑏 ⊆ ∪ 𝑆)
12		nnon 7566	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
13		omsson 7564	. . . . . . . . . . . . . . 15 ⊢ ω ⊆ On
14	13, 10	sseldi 3913	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ On)
15		ontri1 6193	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ On ∧ ∪ 𝑆 ∈ On) → (𝑏 ⊆ ∪ 𝑆 ↔ ¬ ∪ 𝑆 ∈ 𝑏))
16	12, 14, 15	syl2anr 599	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → (𝑏 ⊆ ∪ 𝑆 ↔ ¬ ∪ 𝑆 ∈ 𝑏))
17	11, 16	syl5ib 247	. . . . . . . . . . . 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 458	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → ¬ 𝑏 ∈ 𝑆)
20	6	eleq2i 2881	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝑆 ↔ 𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)})
21	19, 20	sylnib 331	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ (𝑏 ∈ ω ∧ ∪ 𝑆 ∈ 𝑏)) → ¬ 𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)})
22		suceq 6224	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
23	22	fveq2d 6649	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘suc 𝑦) = (𝐹‘suc 𝑏))
24		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
25	23, 24	psseq12d 4022	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → ((𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦) ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
26	25	elrab3 3629	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → (𝑏 ∈ {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} ↔ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
27	26	ad2antrl 727	. . . . . . . . 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 327	. . . . . . . 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 221	. . . . . 6 ⊢ (((𝜑 ∧ 𝑆 ∈ Fin) ∧ 𝑏 ∈ ω) → ¬ (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
32	31	nrexdv 3229	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ¬ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
33		eleq1 2877	. . . . . . . . 9 ⊢ (𝑎 = ∪ 𝑆 → (𝑎 ∈ 𝑏 ↔ ∪ 𝑆 ∈ 𝑏))
34	33	anbi1d 632	. . . . . . . 8 ⊢ (𝑎 = ∪ 𝑆 → ((𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
35	34	rexbidv 3256	. . . . . . 7 ⊢ (𝑎 = ∪ 𝑆 → (∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
36	35	notbid 321	. . . . . 6 ⊢ (𝑎 = ∪ 𝑆 → (¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)) ↔ ¬ ∃𝑏 ∈ ω (∪ 𝑆 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏))))
37	36	rspcev 3571	. . . . 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 587	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ Fin) → ∃𝑎 ∈ ω ¬ ∃𝑏 ∈ ω (𝑎 ∈ 𝑏 ∧ (𝐹‘suc 𝑏) ⊊ (𝐹‘𝑏)))
39		rexnal 3201	. . . 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 221	. . 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 138	1 ⊢ (𝜑 → ¬ 𝑆 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3881   ⊊ wpss 3882  𝒫 cpw 4497  ∪ cuni 4800  ∩ cint 4838  ran crn 5520  Oncon0 6159  suc csuc 6161  ⟶wf 6320  ‘cfv 6324  ωcom 7560  Fincfn 8492

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-er 8272  df-en 8493  df-fin 8496

This theorem is referenced by:  isf32lem6  9769  isf32lem7  9770  isf32lem8  9771