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Theorem zfz1isolemiso 10910

Description: Lemma for zfz1iso 10912. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)

**Hypotheses**
Ref	Expression
zfz1isolemiso.xf	⊢ (𝜑 → 𝑋 ∈ Fin)
zfz1isolemiso.xz	⊢ (𝜑 → 𝑋 ⊆ ℤ)
zfz1isolemiso.mx	⊢ (𝜑 → 𝑀 ∈ 𝑋)
zfz1isolemiso.m	⊢ (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)
zfz1isolemiso.g	⊢ (𝜑 → 𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
zfz1isolemiso.a	⊢ (𝜑 → 𝐴 ∈ (1...(♯‘𝑋)))
zfz1isolemiso.b	⊢ (𝜑 → 𝐵 ∈ (1...(♯‘𝑋)))

**Assertion**
Ref	Expression
zfz1isolemiso	⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))

Distinct variable groups: 𝑧,𝐴 𝑧,𝐵 𝑧,𝐺 𝑧,𝑀 𝑧,𝑋 Allowed substitution hint: 𝜑(𝑧)

**Proof of Theorem zfz1isolemiso**
Step	Hyp	Ref	Expression
1		zfz1isolemiso.g	. . . . . 6 ⊢ (𝜑 → 𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
2	1	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
3		simplr 528	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))))
4		simpr 110	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))))
5		isorel 5851	. . . . 5 ⊢ ((𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) ∧ (𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))))) → (𝐴 < 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵)))
6	2, 3, 4, 5	syl12anc 1247	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐴 < 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵)))
7		zfz1isolemiso.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (1...(♯‘𝑋)))
8	7	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 ∈ (1...(♯‘𝑋)))
9		elfzelz 10091	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (1...(♯‘𝑋)) → 𝐴 ∈ ℤ)
10	7, 9	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℤ)
11	10	zred 9439	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 ∈ ℝ)
13		zfz1isolemiso.xf	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ Fin)
14		hashcl 10852	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
15	13, 14	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ₀)
16	15	nn0red 9294	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ)
17		peano2rem 8286	. . . . . . . . . . 11 ⊢ ((♯‘𝑋) ∈ ℝ → ((♯‘𝑋) − 1) ∈ ℝ)
18	16, 17	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑋) − 1) ∈ ℝ)
19	18	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) ∈ ℝ)
20	16	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ∈ ℝ)
21		elfzle2 10094	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) → 𝐴 ≤ (♯‘(𝑋 ∖ {𝑀})))
22	21	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 ≤ (♯‘(𝑋 ∖ {𝑀})))
23		zfz1isolemiso.mx	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ 𝑋)
24		hashdifsn 10890	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
25	13, 23, 24	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
26	25	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
27	22, 26	breqtrd 4055	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 ≤ ((♯‘𝑋) − 1))
28	20	ltm1d 8951	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) < (♯‘𝑋))
29	12, 19, 20, 27, 28	lelttrd 8144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 < (♯‘𝑋))
30	12, 29	gtned 8132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ≠ 𝐴)
31		fvunsng 5752	. . . . . . 7 ⊢ ((𝐴 ∈ (1...(♯‘𝑋)) ∧ (♯‘𝑋) ≠ 𝐴) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = (𝐺‘𝐴))
32	8, 30, 31	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = (𝐺‘𝐴))
33	32	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = (𝐺‘𝐴))
34		zfz1isolemiso.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (1...(♯‘𝑋)))
35	34	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ (1...(♯‘𝑋)))
36		elfzelz 10091	. . . . . . . . . 10 ⊢ (𝐵 ∈ (1...(♯‘𝑋)) → 𝐵 ∈ ℤ)
37	34, 36	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℤ)
38	37	zred 9439	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
39	38	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ ℝ)
40	18	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) ∈ ℝ)
41	16	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ∈ ℝ)
42		elfzle2 10094	. . . . . . . . . 10 ⊢ (𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) → 𝐵 ≤ (♯‘(𝑋 ∖ {𝑀})))
43	42	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ (♯‘(𝑋 ∖ {𝑀})))
44	25	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
45	43, 44	breqtrd 4055	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ ((♯‘𝑋) − 1))
46	16	ltm1d 8951	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) − 1) < (♯‘𝑋))
47	46	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) < (♯‘𝑋))
48	39, 40, 41, 45, 47	lelttrd 8144	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 < (♯‘𝑋))
49	39, 48	gtned 8132	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ≠ 𝐵)
50		fvunsng 5752	. . . . . 6 ⊢ ((𝐵 ∈ (1...(♯‘𝑋)) ∧ (♯‘𝑋) ≠ 𝐵) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = (𝐺‘𝐵))
51	35, 49, 50	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = (𝐺‘𝐵))
52	33, 51	breq12d 4042	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) ↔ (𝐺‘𝐴) < (𝐺‘𝐵)))
53	6, 52	bitr4d 191	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
54	29	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐴 < (♯‘𝑋))
55		elsni 3636	. . . . . 6 ⊢ (𝐵 ∈ {(♯‘𝑋)} → 𝐵 = (♯‘𝑋))
56	55	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐵 = (♯‘𝑋))
57	54, 56	breqtrrd 4057	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐴 < 𝐵)
58		isof1o 5850	. . . . . . . . . . . . 13 ⊢ (𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) → 𝐺:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}))
59	1, 58	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}))
60		f1of 5500	. . . . . . . . . . . 12 ⊢ (𝐺:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}) → 𝐺:(1...(♯‘(𝑋 ∖ {𝑀})))⟶(𝑋 ∖ {𝑀}))
61	59, 60	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:(1...(♯‘(𝑋 ∖ {𝑀})))⟶(𝑋 ∖ {𝑀}))
62	61	ffvelcdmda 5693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐴) ∈ (𝑋 ∖ {𝑀}))
63	62	eldifbd 3165	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ (𝐺‘𝐴) ∈ {𝑀})
64		elsn2g 3651	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝑋 → ((𝐺‘𝐴) ∈ {𝑀} ↔ (𝐺‘𝐴) = 𝑀))
65	23, 64	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐺‘𝐴) ∈ {𝑀} ↔ (𝐺‘𝐴) = 𝑀))
66	65	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺‘𝐴) ∈ {𝑀} ↔ (𝐺‘𝐴) = 𝑀))
67	63, 66	mtbid 673	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ (𝐺‘𝐴) = 𝑀)
68		breq1 4032	. . . . . . . . . 10 ⊢ (𝑧 = (𝐺‘𝐴) → (𝑧 ≤ 𝑀 ↔ (𝐺‘𝐴) ≤ 𝑀))
69		zfz1isolemiso.m	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)
70	69	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)
71	62	eldifad 3164	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐴) ∈ 𝑋)
72	68, 70, 71	rspcdva 2869	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐴) ≤ 𝑀)
73		zfz1isolemiso.xz	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ⊆ ℤ)
74	73	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝑋 ⊆ ℤ)
75	74, 71	sseldd 3180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐴) ∈ ℤ)
76	73, 23	sseldd 3180	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
77	76	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝑀 ∈ ℤ)
78		zleloe 9364	. . . . . . . . . 10 ⊢ (((𝐺‘𝐴) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝐺‘𝐴) ≤ 𝑀 ↔ ((𝐺‘𝐴) < 𝑀 ∨ (𝐺‘𝐴) = 𝑀)))
79	75, 77, 78	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺‘𝐴) ≤ 𝑀 ↔ ((𝐺‘𝐴) < 𝑀 ∨ (𝐺‘𝐴) = 𝑀)))
80	72, 79	mpbid 147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺‘𝐴) < 𝑀 ∨ (𝐺‘𝐴) = 𝑀))
81	67, 80	ecased 1360	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐴) < 𝑀)
82	15	nn0zd 9437	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑋) ∈ ℤ)
83		peano2zm 9355	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑋) ∈ ℤ → ((♯‘𝑋) − 1) ∈ ℤ)
84	82, 83	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘𝑋) − 1) ∈ ℤ)
85		zltnle 9363	. . . . . . . . . . . . . 14 ⊢ ((((♯‘𝑋) − 1) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → (((♯‘𝑋) − 1) < (♯‘𝑋) ↔ ¬ (♯‘𝑋) ≤ ((♯‘𝑋) − 1)))
86	84, 82, 85	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((♯‘𝑋) − 1) < (♯‘𝑋) ↔ ¬ (♯‘𝑋) ≤ ((♯‘𝑋) − 1)))
87	46, 86	mpbid 147	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (♯‘𝑋) ≤ ((♯‘𝑋) − 1))
88	25	breq2d 4041	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑋) ≤ (♯‘(𝑋 ∖ {𝑀})) ↔ (♯‘𝑋) ≤ ((♯‘𝑋) − 1)))
89	87, 88	mtbird 674	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (♯‘𝑋) ≤ (♯‘(𝑋 ∖ {𝑀})))
90		elfzle2 10094	. . . . . . . . . . 11 ⊢ ((♯‘𝑋) ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) → (♯‘𝑋) ≤ (♯‘(𝑋 ∖ {𝑀})))
91	89, 90	nsyl 629	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (♯‘𝑋) ∈ (1...(♯‘(𝑋 ∖ {𝑀}))))
92		fdm 5409	. . . . . . . . . . 11 ⊢ (𝐺:(1...(♯‘(𝑋 ∖ {𝑀})))⟶(𝑋 ∖ {𝑀}) → dom 𝐺 = (1...(♯‘(𝑋 ∖ {𝑀}))))
93	61, 92	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐺 = (1...(♯‘(𝑋 ∖ {𝑀}))))
94	91, 93	neleqtrrd 2292	. . . . . . . . 9 ⊢ (𝜑 → ¬ (♯‘𝑋) ∈ dom 𝐺)
95		fsnunfv 5759	. . . . . . . . 9 ⊢ (((♯‘𝑋) ∈ ℕ₀ ∧ 𝑀 ∈ 𝑋 ∧ ¬ (♯‘𝑋) ∈ dom 𝐺) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)) = 𝑀)
96	15, 23, 94, 95	syl3anc 1249	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)) = 𝑀)
97	96	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)) = 𝑀)
98	81, 32, 97	3brtr4d 4061	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
99	98	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
100	56	fveq2d 5558	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
101	99, 100	breqtrrd 4057	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵))
102	57, 101	2thd 175	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
103	13, 23	zfz1isolemsplit 10909	. . . . . 6 ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
104	34, 103	eleqtrd 2272	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
105		elun 3300	. . . . 5 ⊢ (𝐵 ∈ ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}) ↔ (𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) ∨ 𝐵 ∈ {(♯‘𝑋)}))
106	104, 105	sylib 122	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) ∨ 𝐵 ∈ {(♯‘𝑋)}))
107	106	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) ∨ 𝐵 ∈ {(♯‘𝑋)}))
108	53, 102, 107	mpjaodan 799	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
109	38	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ ℝ)
110	11	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 ∈ ℝ)
111		elfzle2 10094	. . . . . . . 8 ⊢ (𝐵 ∈ (1...(♯‘𝑋)) → 𝐵 ≤ (♯‘𝑋))
112	34, 111	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ (♯‘𝑋))
113	112	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ (♯‘𝑋))
114		elsni 3636	. . . . . . 7 ⊢ (𝐴 ∈ {(♯‘𝑋)} → 𝐴 = (♯‘𝑋))
115	114	ad2antlr 489	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 = (♯‘𝑋))
116	113, 115	breqtrrd 4057	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ 𝐴)
117	109, 110, 116	lensymd 8141	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ 𝐴 < 𝐵)
118	73	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝑋 ⊆ ℤ)
119	61	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐺:(1...(♯‘(𝑋 ∖ {𝑀})))⟶(𝑋 ∖ {𝑀}))
120		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))))
121	119, 120	ffvelcdmd 5694	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ∈ (𝑋 ∖ {𝑀}))
122	121	eldifad 3164	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ∈ 𝑋)
123	118, 122	sseldd 3180	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ∈ ℤ)
124	123	zred 9439	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ∈ ℝ)
125	76	zred 9439	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
126	125	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝑀 ∈ ℝ)
127		breq1 4032	. . . . . . 7 ⊢ (𝑧 = (𝐺‘𝐵) → (𝑧 ≤ 𝑀 ↔ (𝐺‘𝐵) ≤ 𝑀))
128	69	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)
129	127, 128, 122	rspcdva 2869	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ≤ 𝑀)
130	124, 126, 129	lensymd 8141	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ 𝑀 < (𝐺‘𝐵))
131	115	fveq2d 5558	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
132	96	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)) = 𝑀)
133	131, 132	eqtrd 2226	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = 𝑀)
134	34	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ (1...(♯‘𝑋)))
135	18	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) ∈ ℝ)
136	16	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ∈ ℝ)
137	42	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ (♯‘(𝑋 ∖ {𝑀})))
138	25	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
139	137, 138	breqtrd 4055	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ ((♯‘𝑋) − 1))
140	46	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) < (♯‘𝑋))
141	109, 135, 136, 139, 140	lelttrd 8144	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 < (♯‘𝑋))
142	109, 141	gtned 8132	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ≠ 𝐵)
143	134, 142, 50	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = (𝐺‘𝐵))
144	133, 143	breq12d 4042	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) ↔ 𝑀 < (𝐺‘𝐵)))
145	130, 144	mtbird 674	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵))
146	117, 145	2falsed 703	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
147	38	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐵 ∈ ℝ)
148	147	ltnrd 8131	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ¬ 𝐵 < 𝐵)
149	114	ad2antlr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐴 = (♯‘𝑋))
150	55	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐵 = (♯‘𝑋))
151	149, 150	eqtr4d 2229	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐴 = 𝐵)
152	151	breq1d 4039	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → (𝐴 < 𝐵 ↔ 𝐵 < 𝐵))
153	148, 152	mtbird 674	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ¬ 𝐴 < 𝐵)
154	125	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝑀 ∈ ℝ)
155	154	ltnrd 8131	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ¬ 𝑀 < 𝑀)
156	149	fveq2d 5558	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
157	96	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)) = 𝑀)
158	156, 157	eqtrd 2226	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = 𝑀)
159	150	fveq2d 5558	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
160	159, 157	eqtrd 2226	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = 𝑀)
161	158, 160	breq12d 4042	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → (((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) ↔ 𝑀 < 𝑀))
162	155, 161	mtbird 674	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ¬ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵))
163	153, 162	2falsed 703	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
164	106	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) → (𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) ∨ 𝐵 ∈ {(♯‘𝑋)}))
165	146, 163, 164	mpjaodan 799	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
166	7, 103	eleqtrd 2272	. . 3 ⊢ (𝜑 → 𝐴 ∈ ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
167		elun 3300	. . 3 ⊢ (𝐴 ∈ ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}) ↔ (𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) ∨ 𝐴 ∈ {(♯‘𝑋)}))
168	166, 167	sylib 122	. 2 ⊢ (𝜑 → (𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) ∨ 𝐴 ∈ {(♯‘𝑋)}))
169	108, 165, 168	mpjaodan 799	1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 ∀wral 2472 ∖ cdif 3150 ∪ cun 3151 ⊆ wss 3153 {csn 3618 ⟨cop 3621 class class class wbr 4029 dom cdm 4659 ⟶wf 5250 –1-1-onto→wf1o 5253 ‘cfv 5254 Isom wiso 5255 (class class class)co 5918 Fincfn 6794 ℝcr 7871 1c1 7873 < clt 8054 ≤ cle 8055 − cmin 8190 ℕ₀cn0 9240 ℤcz 9317 ...cfz 10074 ♯chash 10846

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-addass 7974 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-0id 7980 ax-rnegex 7981 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-inn 8983 df-n0 9241 df-z 9318 df-uz 9593 df-fz 10075 df-ihash 10847

This theorem is referenced by: zfz1isolem1 10911

W3C validator