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Theorem zfz1isolemiso 11314

Description: Lemma for zfz1iso 11316. 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 492	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
3		simplr 533	. . . . 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 6017	. . . . 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 1276	. . . 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 10446	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (1...(♯‘𝑋)) → 𝐴 ∈ ℤ)
10	7, 9	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℤ)
11	10	zred 9780	. . . . . . . . 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 11243	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
15	13, 14	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ₀)
16	15	nn0red 9633	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ)
17		peano2rem 8601	. . . . . . . . . . 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 10450	. . . . . . . . . . 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 11283	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
25	13, 23, 24	syl2anc 415	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
26	25	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
27	22, 26	breqtrd 4157	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 ≤ ((♯‘𝑋) − 1))
28	20	ltm1d 9271	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) < (♯‘𝑋))
29	12, 19, 20, 27, 28	lelttrd 8459	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 < (♯‘𝑋))
30	12, 29	gtned 8446	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ≠ 𝐴)
31		fvunsng 5912	. . . . . . 7 ⊢ ((𝐴 ∈ (1...(♯‘𝑋)) ∧ (♯‘𝑋) ≠ 𝐴) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = (𝐺‘𝐴))
32	8, 30, 31	syl2anc 415	. . . . . 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 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ (1...(♯‘𝑋)))
36		elfzelz 10446	. . . . . . . . . 10 ⊢ (𝐵 ∈ (1...(♯‘𝑋)) → 𝐵 ∈ ℤ)
37	34, 36	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℤ)
38	37	zred 9780	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
39	38	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ ℝ)
40	18	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) ∈ ℝ)
41	16	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ∈ ℝ)
42		elfzle2 10450	. . . . . . . . . 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 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
45	43, 44	breqtrd 4157	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ ((♯‘𝑋) − 1))
46	16	ltm1d 9271	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) − 1) < (♯‘𝑋))
47	46	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) < (♯‘𝑋))
48	39, 40, 41, 45, 47	lelttrd 8459	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 < (♯‘𝑋))
49	39, 48	gtned 8446	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ≠ 𝐵)
50		fvunsng 5912	. . . . . 6 ⊢ ((𝐵 ∈ (1...(♯‘𝑋)) ∧ (♯‘𝑋) ≠ 𝐵) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = (𝐺‘𝐵))
51	35, 49, 50	syl2anc 415	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = (𝐺‘𝐵))
52	33, 51	breq12d 4144	. . . 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 3727	. . . . . 6 ⊢ (𝐵 ∈ {(♯‘𝑋)} → 𝐵 = (♯‘𝑋))
56	55	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐵 = (♯‘𝑋))
57	54, 56	breqtrrd 4159	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐴 < 𝐵)
58		isof1o 6016	. . . . . . . . . . . . 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 5642	. . . . . . . . . . . 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 5846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐴) ∈ (𝑋 ∖ {𝑀}))
63	62	eldifbd 3232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ (𝐺‘𝐴) ∈ {𝑀})
64		elsn2g 3742	. . . . . . . . . . 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 683	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ (𝐺‘𝐴) = 𝑀)
68		breq1 4134	. . . . . . . . . 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 3231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐴) ∈ 𝑋)
72	68, 70, 71	rspcdva 2934	. . . . . . . . 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 3249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐴) ∈ ℤ)
76	73, 23	sseldd 3249	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
77	76	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝑀 ∈ ℤ)
78		zleloe 9703	. . . . . . . . . 10 ⊢ (((𝐺‘𝐴) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝐺‘𝐴) ≤ 𝑀 ↔ ((𝐺‘𝐴) < 𝑀 ∨ (𝐺‘𝐴) = 𝑀)))
79	75, 77, 78	syl2anc 415	. . . . . . . . 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 1390	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐴) < 𝑀)
82	15	nn0zd 9778	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑋) ∈ ℤ)
83		peano2zm 9694	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑋) ∈ ℤ → ((♯‘𝑋) − 1) ∈ ℤ)
84	82, 83	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘𝑋) − 1) ∈ ℤ)
85		zltnle 9702	. . . . . . . . . . . . . 14 ⊢ ((((♯‘𝑋) − 1) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → (((♯‘𝑋) − 1) < (♯‘𝑋) ↔ ¬ (♯‘𝑋) ≤ ((♯‘𝑋) − 1)))
86	84, 82, 85	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((♯‘𝑋) − 1) < (♯‘𝑋) ↔ ¬ (♯‘𝑋) ≤ ((♯‘𝑋) − 1)))
87	46, 86	mpbid 147	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (♯‘𝑋) ≤ ((♯‘𝑋) − 1))
88	25	breq2d 4143	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑋) ≤ (♯‘(𝑋 ∖ {𝑀})) ↔ (♯‘𝑋) ≤ ((♯‘𝑋) − 1)))
89	87, 88	mtbird 684	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (♯‘𝑋) ≤ (♯‘(𝑋 ∖ {𝑀})))
90		elfzle2 10450	. . . . . . . . . . 11 ⊢ ((♯‘𝑋) ∈ (1...(♯‘(𝑋 ∖ {𝑀}))) → (♯‘𝑋) ≤ (♯‘(𝑋 ∖ {𝑀})))
91	89, 90	nsyl 637	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (♯‘𝑋) ∈ (1...(♯‘(𝑋 ∖ {𝑀}))))
92		fdm 5542	. . . . . . . . . . 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 2337	. . . . . . . . 9 ⊢ (𝜑 → ¬ (♯‘𝑋) ∈ dom 𝐺)
95		fsnunfv 5919	. . . . . . . . 9 ⊢ (((♯‘𝑋) ∈ ℕ₀ ∧ 𝑀 ∈ 𝑋 ∧ ¬ (♯‘𝑋) ∈ dom 𝐺) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)) = 𝑀)
96	15, 23, 94, 95	syl3anc 1278	. . . . . . . 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 4163	. . . . . 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 5702	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
101	99, 100	breqtrrd 4159	. . . 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 11313	. . . . . 6 ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
104	34, 103	eleqtrd 2317	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
105		elun 3370	. . . . 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 810	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
109	38	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ ℝ)
110	11	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 ∈ ℝ)
111		elfzle2 10450	. . . . . . . 8 ⊢ (𝐵 ∈ (1...(♯‘𝑋)) → 𝐵 ≤ (♯‘𝑋))
112	34, 111	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ (♯‘𝑋))
113	112	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ (♯‘𝑋))
114		elsni 3727	. . . . . . 7 ⊢ (𝐴 ∈ {(♯‘𝑋)} → 𝐴 = (♯‘𝑋))
115	114	ad2antlr 493	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐴 = (♯‘𝑋))
116	113, 115	breqtrrd 4159	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ 𝐴)
117	109, 110, 116	lensymd 8456	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ 𝐴 < 𝐵)
118	73	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝑋 ⊆ ℤ)
119	61	ad2antrr 492	. . . . . . . . . 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 5847	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ∈ (𝑋 ∖ {𝑀}))
122	121	eldifad 3231	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ∈ 𝑋)
123	118, 122	sseldd 3249	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ∈ ℤ)
124	123	zred 9780	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ∈ ℝ)
125	76	zred 9780	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
126	125	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝑀 ∈ ℝ)
127		breq1 4134	. . . . . . 7 ⊢ (𝑧 = (𝐺‘𝐵) → (𝑧 ≤ 𝑀 ↔ (𝐺‘𝐵) ≤ 𝑀))
128	69	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)
129	127, 128, 122	rspcdva 2934	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐺‘𝐵) ≤ 𝑀)
130	124, 126, 129	lensymd 8456	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ 𝑀 < (𝐺‘𝐵))
131	115	fveq2d 5702	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
132	96	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)) = 𝑀)
133	131, 132	eqtrd 2271	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = 𝑀)
134	34	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ∈ (1...(♯‘𝑋)))
135	18	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) ∈ ℝ)
136	16	ad2antrr 492	. . . . . . . . 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 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
139	137, 138	breqtrd 4157	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 ≤ ((♯‘𝑋) − 1))
140	46	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((♯‘𝑋) − 1) < (♯‘𝑋))
141	109, 135, 136, 139, 140	lelttrd 8459	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → 𝐵 < (♯‘𝑋))
142	109, 141	gtned 8446	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (♯‘𝑋) ≠ 𝐵)
143	134, 142, 50	syl2anc 415	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = (𝐺‘𝐵))
144	133, 143	breq12d 4144	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) ↔ 𝑀 < (𝐺‘𝐵)))
145	130, 144	mtbird 684	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → ¬ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵))
146	117, 145	2falsed 714	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))) → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
147	38	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐵 ∈ ℝ)
148	147	ltnrd 8445	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ¬ 𝐵 < 𝐵)
149	114	ad2antlr 493	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐴 = (♯‘𝑋))
150	55	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐵 = (♯‘𝑋))
151	149, 150	eqtr4d 2274	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝐴 = 𝐵)
152	151	breq1d 4141	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → (𝐴 < 𝐵 ↔ 𝐵 < 𝐵))
153	148, 152	mtbird 684	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ¬ 𝐴 < 𝐵)
154	125	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → 𝑀 ∈ ℝ)
155	154	ltnrd 8445	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ¬ 𝑀 < 𝑀)
156	149	fveq2d 5702	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
157	96	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)) = 𝑀)
158	156, 157	eqtrd 2271	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) = 𝑀)
159	150	fveq2d 5702	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘(♯‘𝑋)))
160	159, 157	eqtrd 2271	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) = 𝑀)
161	158, 160	breq12d 4144	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → (((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵) ↔ 𝑀 < 𝑀))
162	155, 161	mtbird 684	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) ∧ 𝐵 ∈ {(♯‘𝑋)}) → ¬ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵))
163	153, 162	2falsed 714	. . 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 810	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ {(♯‘𝑋)}) → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))
166	7, 103	eleqtrd 2317	. . 3 ⊢ (𝜑 → 𝐴 ∈ ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
167		elun 3370	. . 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 810	1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∀wral 2528 ∖ cdif 3217 ∪ cun 3218 ⊆ wss 3220 {csn 3709 ⟨cop 3712 class class class wbr 4131 dom cdm 4776 ⟶wf 5376 –1-1-onto→wf1o 5379 ‘cfv 5380 Isom wiso 5381 (class class class)co 6088 Fincfn 7027 ℝcr 8185 1c1 8187 < clt 8367 ≤ cle 8368 − cmin 8505 ℕ₀cn0 9575 ℤcz 9656 ...cfz 10429 ♯chash 11237

This proof depends on 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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-addcom 8286 ax-addass 8288 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-0id 8294 ax-rnegex 8295 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302

This proof 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 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-en 7028 df-dom 7029 df-fin 7030 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-inn 9315 df-n0 9576 df-z 9657 df-uz 9939 df-fz 10430 df-ihash 11238

This theorem is used by: zfz1isolem1 11315

W3C validator