Theorem ennnfonelemrnh 12902

Description: Lemma for ennnfone 12911. A consequence of ennnfonelemss 12896. (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 12890	. . . . 5 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
9	8	ffund 5449	. . . 4 ⊢ (𝜑 → Fun 𝐻)
10		ennnfonelemrnh.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐻)
11		elrnrexdm 5742	. . . 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 5452	. . . 4 ⊢ (𝜑 → dom 𝐻 = ℕ0)
14	13	rexeqdv 2712	. . 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 5742	. . . . . 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 2712	. . . . 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 9528	. . . . . 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 9528	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑡 ∈ ℤ)
26		zletric 9451	. . . . . 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 12897	. . . . . . 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 12897	. . . . . . 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 788	. . . . 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 3232	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑋 ⊆ 𝑌 ↔ (𝐻‘𝑠) ⊆ (𝐻‘𝑡)))
49	47, 46	sseq12d 3232	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑌 ⊆ 𝑋 ↔ (𝐻‘𝑡) ⊆ (𝐻‘𝑠)))
50	48, 49	orbi12d 795	. . . 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 2630	. 2 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))
53	15, 52	rexlimddv 2630	1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 710  DECID wdc 836   = wceq 1373   ∈ wcel 2178   ≠ wne 2378  ∀wral 2486  ∃wrex 2487   ∪ cun 3172   ⊆ wss 3174  ∅c0 3468  ifcif 3579  {csn 3643  ⟨cop 3646   class class class wbr 4059   ↦ cmpt 4121  suc csuc 4430  ωcom 4656  ^◡ccnv 4692  dom cdm 4693  ran crn 4694   “ cima 4696  Fun wfun 5284  –onto→wfo 5288  ‘cfv 5290  (class class class)co 5967   ∈ cmpo 5969  freccfrec 6499   ↑_pm cpm 6759  0cc0 7960  1c1 7961   + caddc 7963   ≤ cle 8143   − cmin 8278  ℕ₀cn0 9330  ℤcz 9407  seqcseq 10629

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pm 6761  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-seqfrec 10630

This theorem is referenced by:  ennnfonelemfun  12903  ennnfonelemf1  12904