Theorem ennnfonelemrnh 12658

Description: Lemma for ennnfone 12667. A consequence of ennnfonelemss 12652. (Contributed by Jim Kingdon, 16-Jul-2023.)

Hypotheses

Ref	Expression
ennnfonelemh.dceq	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f	⊢ (𝜑 → 𝐹:ω–onto→𝐴)
ennnfonelemh.ne	⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
ennnfonelemh.g	⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
ennnfonelemh.n	⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ennnfonelemh.j	⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
ennnfonelemh.h	⊢ 𝐻 = seq0(𝐺, 𝐽)
ennnfonelemrnh.x	⊢ (𝜑 → 𝑋 ∈ ran 𝐻)
ennnfonelemrnh.y	⊢ (𝜑 → 𝑌 ∈ ran 𝐻)

Assertion

Ref	Expression
ennnfonelemrnh	⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))

Distinct variable groups:   𝐴,𝑗,𝑥,𝑦   𝑗,𝐹,𝑘,𝑛   𝑥,𝐹,𝑦   𝑗,𝐺   𝑥,𝐻,𝑦   𝑗,𝐽   𝑥,𝑁,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑗,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑘,𝑛)   𝐴(𝑘,𝑛)   𝐺(𝑥,𝑦,𝑘,𝑛)   𝐻(𝑗,𝑘,𝑛)   𝐽(𝑥,𝑦,𝑘,𝑛)   𝑁(𝑗,𝑘,𝑛)   𝑋(𝑗,𝑘,𝑛)   𝑌(𝑗,𝑘,𝑛)

Proof of Theorem ennnfonelemrnh

Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemh.dceq	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
2		ennnfonelemh.f	. . . . . 6 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
3		ennnfonelemh.ne	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
4		ennnfonelemh.g	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
5		ennnfonelemh.n	. . . . . 6 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
6		ennnfonelemh.j	. . . . . 6 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
7		ennnfonelemh.h	. . . . . 6 ⊢ 𝐻 = seq0(𝐺, 𝐽)
8	1, 2, 3, 4, 5, 6, 7	ennnfonelemh 12646	. . . . 5 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
9	8	ffund 5414	. . . 4 ⊢ (𝜑 → Fun 𝐻)
10		ennnfonelemrnh.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐻)
11		elrnrexdm 5704	. . . 4 ⊢ (Fun 𝐻 → (𝑋 ∈ ran 𝐻 → ∃𝑠 ∈ dom 𝐻 𝑋 = (𝐻‘𝑠)))
12	9, 10, 11	sylc 62	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ dom 𝐻 𝑋 = (𝐻‘𝑠))
13	8	fdmd 5417	. . . 4 ⊢ (𝜑 → dom 𝐻 = ℕ0)
14	13	rexeqdv 2700	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ dom 𝐻 𝑋 = (𝐻‘𝑠) ↔ ∃𝑠 ∈ ℕ0 𝑋 = (𝐻‘𝑠)))
15	12, 14	mpbid 147	. 2 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 𝑋 = (𝐻‘𝑠))
16		ennnfonelemrnh.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ran 𝐻)
17		elrnrexdm 5704	. . . . . 6 ⊢ (Fun 𝐻 → (𝑌 ∈ ran 𝐻 → ∃𝑡 ∈ dom 𝐻 𝑌 = (𝐻‘𝑡)))
18	9, 16, 17	sylc 62	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ dom 𝐻 𝑌 = (𝐻‘𝑡))
19	13	rexeqdv 2700	. . . . 5 ⊢ (𝜑 → (∃𝑡 ∈ dom 𝐻 𝑌 = (𝐻‘𝑡) ↔ ∃𝑡 ∈ ℕ0 𝑌 = (𝐻‘𝑡)))
20	18, 19	mpbid 147	. . . 4 ⊢ (𝜑 → ∃𝑡 ∈ ℕ0 𝑌 = (𝐻‘𝑡))
21	20	adantr 276	. . 3 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) → ∃𝑡 ∈ ℕ0 𝑌 = (𝐻‘𝑡))
22		simplrl 535	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑠 ∈ ℕ0)
23	22	nn0zd 9463	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑠 ∈ ℤ)
24		simprl 529	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑡 ∈ ℕ0)
25	24	nn0zd 9463	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑡 ∈ ℤ)
26		zletric 9387	. . . . . 6 ⊢ ((𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑠 ≤ 𝑡 ∨ 𝑡 ≤ 𝑠))
27	23, 25, 26	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑠 ≤ 𝑡 ∨ 𝑡 ≤ 𝑠))
28	1	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑠 ≤ 𝑡) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
29	2	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑠 ≤ 𝑡) → 𝐹:ω–onto→𝐴)
30	3	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑠 ≤ 𝑡) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
31	22	adantr 276	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑠 ≤ 𝑡) → 𝑠 ∈ ℕ0)
32		simplrl 535	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑠 ≤ 𝑡) → 𝑡 ∈ ℕ0)
33		simpr 110	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑠 ≤ 𝑡) → 𝑠 ≤ 𝑡)
34	28, 29, 30, 4, 5, 6, 7, 31, 32, 33	ennnfoneleminc 12653	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑠 ≤ 𝑡) → (𝐻‘𝑠) ⊆ (𝐻‘𝑡))
35	34	ex 115	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑠 ≤ 𝑡 → (𝐻‘𝑠) ⊆ (𝐻‘𝑡)))
36	1	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑡 ≤ 𝑠) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
37	2	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑡 ≤ 𝑠) → 𝐹:ω–onto→𝐴)
38	3	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑡 ≤ 𝑠) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
39		simplrl 535	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑡 ≤ 𝑠) → 𝑡 ∈ ℕ0)
40	22	adantr 276	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑡 ≤ 𝑠) → 𝑠 ∈ ℕ0)
41		simpr 110	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑡 ≤ 𝑠) → 𝑡 ≤ 𝑠)
42	36, 37, 38, 4, 5, 6, 7, 39, 40, 41	ennnfoneleminc 12653	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑡 ≤ 𝑠) → (𝐻‘𝑡) ⊆ (𝐻‘𝑠))
43	42	ex 115	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑡 ≤ 𝑠 → (𝐻‘𝑡) ⊆ (𝐻‘𝑠)))
44	35, 43	orim12d 787	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → ((𝑠 ≤ 𝑡 ∨ 𝑡 ≤ 𝑠) → ((𝐻‘𝑠) ⊆ (𝐻‘𝑡) ∨ (𝐻‘𝑡) ⊆ (𝐻‘𝑠))))
45	27, 44	mpd 13	. . . 4 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → ((𝐻‘𝑠) ⊆ (𝐻‘𝑡) ∨ (𝐻‘𝑡) ⊆ (𝐻‘𝑠)))
46		simplrr 536	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑋 = (𝐻‘𝑠))
47		simprr 531	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑌 = (𝐻‘𝑡))
48	46, 47	sseq12d 3215	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑋 ⊆ 𝑌 ↔ (𝐻‘𝑠) ⊆ (𝐻‘𝑡)))
49	47, 46	sseq12d 3215	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑌 ⊆ 𝑋 ↔ (𝐻‘𝑡) ⊆ (𝐻‘𝑠)))
50	48, 49	orbi12d 794	. . . 4 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → ((𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋) ↔ ((𝐻‘𝑠) ⊆ (𝐻‘𝑡) ∨ (𝐻‘𝑡) ⊆ (𝐻‘𝑠))))
51	45, 50	mpbird 167	. . 3 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))
52	21, 51	rexlimddv 2619	. 2 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))
53	15, 52	rexlimddv 2619	1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  ∃wrex 2476   ∪ cun 3155   ⊆ wss 3157  ∅c0 3451  ifcif 3562  {csn 3623  ⟨cop 3626   class class class wbr 4034   ↦ cmpt 4095  suc csuc 4401  ωcom 4627  ^◡ccnv 4663  dom cdm 4664  ran crn 4665   “ cima 4667  Fun wfun 5253  –onto→wfo 5257  ‘cfv 5259  (class class class)co 5925   ∈ cmpo 5927  freccfrec 6457   ↑_pm cpm 6717  0cc0 7896  1c1 7897   + caddc 7899   ≤ cle 8079   − cmin 8214  ℕ₀cn0 9266  ℤcz 9343  seqcseq 10556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pm 6719  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-seqfrec 10557

This theorem is referenced by:  ennnfonelemfun  12659  ennnfonelemf1  12660