Theorem ennnfonelemrnh 13285

Description: Lemma for ennnfone 13294. A consequence of ennnfonelemss 13279. (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 13273	. . . . 5 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
9	8	ffund 5532	. . . 4 ⊢ (𝜑 → Fun 𝐻)
10		ennnfonelemrnh.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐻)
11		elrnrexdm 5838	. . . 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 5535	. . . 4 ⊢ (𝜑 → dom 𝐻 = ℕ0)
14	13	rexeqdv 2756	. . 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 5838	. . . . . 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 2756	. . . . 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 541	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑠 ∈ ℕ0)
23	22	nn0zd 9745	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑠 ∈ ℤ)
24		simprl 535	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑡 ∈ ℕ0)
25	24	nn0zd 9745	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑡 ∈ ℤ)
26		zletric 9667	. . . . . 6 ⊢ ((𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑠 ≤ 𝑡 ∨ 𝑡 ≤ 𝑠))
27	23, 25, 26	syl2anc 415	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑠 ≤ 𝑡 ∨ 𝑡 ≤ 𝑠))
28	1	ad3antrrr 496	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑠 ≤ 𝑡) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
29	2	ad3antrrr 496	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑠 ≤ 𝑡) → 𝐹:ω–onto→𝐴)
30	3	ad3antrrr 496	. . . . . . . 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 541	. . . . . . . 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 13280	. . . . . . 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 496	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑡 ≤ 𝑠) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
37	2	ad3antrrr 496	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) ∧ 𝑡 ≤ 𝑠) → 𝐹:ω–onto→𝐴)
38	3	ad3antrrr 496	. . . . . . . 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 541	. . . . . . . 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 13280	. . . . . . 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 798	. . . . 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 542	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑋 = (𝐻‘𝑠))
47		simprr 537	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑌 = (𝐻‘𝑡))
48	46, 47	sseq12d 3279	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑋 ⊆ 𝑌 ↔ (𝐻‘𝑠) ⊆ (𝐻‘𝑡)))
49	47, 46	sseq12d 3279	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑌 ⊆ 𝑋 ↔ (𝐻‘𝑡) ⊆ (𝐻‘𝑠)))
50	48, 49	orbi12d 805	. . . 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 2673	. 2 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))
53	15, 52	rexlimddv 2673	1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  ∃wrex 2529   ∪ cun 3218   ⊆ wss 3220  ∅c0 3520  ifcif 3635  {csn 3705  ⟨cop 3708   class class class wbr 4125   ↦ cmpt 4187  suc csuc 4505  ωcom 4732  ^◡ccnv 4768  dom cdm 4769  ran crn 4770   “ cima 4772  Fun wfun 5366  –onto→wfo 5370  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  freccfrec 6651   ↑_pm cpm 6913  0cc0 8169  1c1 8170   + caddc 8172   ≤ cle 8351   − cmin 8487  ℕ₀cn0 9542  ℤcz 9623  seqcseq 10862

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pm 6915  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863

This theorem is referenced by:  ennnfonelemfun  13286  ennnfonelemf1  13287