Theorem ennnfonelemrnh 13184

Description: Lemma for ennnfone 13193. A consequence of ennnfonelemss 13178. (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 13172	. . . . 5 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
9	8	ffund 5514	. . . 4 ⊢ (𝜑 → Fun 𝐻)
10		ennnfonelemrnh.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐻)
11		elrnrexdm 5818	. . . 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 5517	. . . 4 ⊢ (𝜑 → dom 𝐻 = ℕ0)
14	13	rexeqdv 2750	. . 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 5818	. . . . . 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 2750	. . . . 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 537	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑠 ∈ ℕ0)
23	22	nn0zd 9701	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑠 ∈ ℤ)
24		simprl 531	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑡 ∈ ℕ0)
25	24	nn0zd 9701	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑡 ∈ ℤ)
26		zletric 9623	. . . . . 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 537	. . . . . . . 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 13179	. . . . . . 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 537	. . . . . . . 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 13179	. . . . . . 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 794	. . . . 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 538	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑋 = (𝐻‘𝑠))
47		simprr 533	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑌 = (𝐻‘𝑡))
48	46, 47	sseq12d 3271	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑋 ⊆ 𝑌 ↔ (𝐻‘𝑠) ⊆ (𝐻‘𝑡)))
49	47, 46	sseq12d 3271	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑌 ⊆ 𝑋 ↔ (𝐻‘𝑡) ⊆ (𝐻‘𝑠)))
50	48, 49	orbi12d 801	. . . 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 2667	. 2 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))
53	15, 52	rexlimddv 2667	1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2205   ≠ wne 2414  ∀wral 2522  ∃wrex 2523   ∪ cun 3211   ⊆ wss 3213  ∅c0 3510  ifcif 3622  {csn 3691  ⟨cop 3694   class class class wbr 4111   ↦ cmpt 4173  suc csuc 4488  ωcom 4714  ^◡ccnv 4750  dom cdm 4751  ran crn 4752   “ cima 4754  Fun wfun 5348  –onto→wfo 5352  ‘cfv 5354  (class class class)co 6052   ∈ cmpo 6054  freccfrec 6623   ↑_pm cpm 6885  0cc0 8129  1c1 8130   + caddc 8132   ≤ cle 8311   − cmin 8446  ℕ₀cn0 9498  ℤcz 9579  seqcseq 10813

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-pm 6887  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-seqfrec 10814

This theorem is referenced by:  ennnfonelemfun  13185  ennnfonelemf1  13186