Theorem ennnfonelemrnh 12411

Description: Lemma for ennnfone 12420. A consequence of ennnfonelemss 12405. (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 12399	. . . . 5 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
9	8	ffund 5369	. . . 4 ⊢ (𝜑 → Fun 𝐻)
10		ennnfonelemrnh.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐻)
11		elrnrexdm 5655	. . . 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 5372	. . . 4 ⊢ (𝜑 → dom 𝐻 = ℕ0)
14	13	rexeqdv 2679	. . 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 5655	. . . . . 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 2679	. . . . 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 9371	. . . . . 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 9371	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → 𝑡 ∈ ℤ)
26		zletric 9295	. . . . . 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 12406	. . . . . . 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 12406	. . . . . . 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 786	. . . . 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 3186	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑋 ⊆ 𝑌 ↔ (𝐻‘𝑠) ⊆ (𝐻‘𝑡)))
49	47, 46	sseq12d 3186	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) ∧ (𝑡 ∈ ℕ0 ∧ 𝑌 = (𝐻‘𝑡))) → (𝑌 ⊆ 𝑋 ↔ (𝐻‘𝑡) ⊆ (𝐻‘𝑠)))
50	48, 49	orbi12d 793	. . . 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 2599	. 2 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ0 ∧ 𝑋 = (𝐻‘𝑠))) → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))
53	15, 52	rexlimddv 2599	1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∨ 𝑌 ⊆ 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  ∃wrex 2456   ∪ cun 3127   ⊆ wss 3129  ∅c0 3422  ifcif 3534  {csn 3592  ⟨cop 3595   class class class wbr 4003   ↦ cmpt 4064  suc csuc 4365  ωcom 4589  ^◡ccnv 4625  dom cdm 4626  ran crn 4627   “ cima 4629  Fun wfun 5210  –onto→wfo 5214  ‘cfv 5216  (class class class)co 5874   ∈ cmpo 5876  freccfrec 6390   ↑_pm cpm 6648  0cc0 7810  1c1 7811   + caddc 7813   ≤ cle 7991   − cmin 8126  ℕ₀cn0 9174  ℤcz 9251  seqcseq 10442

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pm 6650  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-seqfrec 10443

This theorem is referenced by:  ennnfonelemfun  12412  ennnfonelemf1  12413