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Theorem uzwo 11627

Description: Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by NM, 8-Oct-2005.)

**Assertion**
Ref	Expression
uzwo	⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)

Distinct variable group: 𝑗,𝑘,𝑆 Allowed substitution hints: 𝑀(𝑗,𝑘)

Proof of Theorem uzwo Dummy variables 𝑡 ℎ 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4586	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑀 → (ℎ ≤ 𝑡 ↔ 𝑀 ≤ 𝑡))
2	1	ralbidv 2969	. . . . . . . . . . 11 ⊢ (ℎ = 𝑀 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡))
3	2	imbi2d 329	. . . . . . . . . 10 ⊢ (ℎ = 𝑀 → (((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡) ↔ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡)))
4		breq1 4586	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑚 → (ℎ ≤ 𝑡 ↔ 𝑚 ≤ 𝑡))
5	4	ralbidv 2969	. . . . . . . . . . 11 ⊢ (ℎ = 𝑚 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡))
6	5	imbi2d 329	. . . . . . . . . 10 ⊢ (ℎ = 𝑚 → (((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡) ↔ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡)))
7		breq1 4586	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑚 + 1) → (ℎ ≤ 𝑡 ↔ (𝑚 + 1) ≤ 𝑡))
8	7	ralbidv 2969	. . . . . . . . . . 11 ⊢ (ℎ = (𝑚 + 1) → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))
9	8	imbi2d 329	. . . . . . . . . 10 ⊢ (ℎ = (𝑚 + 1) → (((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡) ↔ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)))
10		breq1 4586	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑛 → (ℎ ≤ 𝑡 ↔ 𝑛 ≤ 𝑡))
11	10	ralbidv 2969	. . . . . . . . . . 11 ⊢ (ℎ = 𝑛 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡))
12	11	imbi2d 329	. . . . . . . . . 10 ⊢ (ℎ = 𝑛 → (((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡) ↔ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡)))
13		ssel 3562	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (𝑡 ∈ 𝑆 → 𝑡 ∈ (ℤ_≥‘𝑀)))
14		eluzle 11576	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑡)
15	13, 14	syl6 34	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (𝑡 ∈ 𝑆 → 𝑀 ≤ 𝑡))
16	15	ralrimiv 2948	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡)
17	16	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡)
18	17	a1i 11	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡))
19		uzssz 11583	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
20		sstr 3576	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ (ℤ_≥‘𝑀) ⊆ ℤ) → 𝑆 ⊆ ℤ)
21	19, 20	mpan2 703	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → 𝑆 ⊆ ℤ)
22		eluzelz 11573	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ℤ_≥‘𝑀) → 𝑚 ∈ ℤ)
23		breq1 4586	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑚 → (𝑗 ≤ 𝑡 ↔ 𝑚 ≤ 𝑡))
24	23	ralbidv 2969	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑚 → (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡))
25	24	rspcev 3282	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)
26	25	expcom 450	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → (𝑚 ∈ 𝑆 → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡))
27	26	con3rr3 150	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ¬ 𝑚 ∈ 𝑆))
28		ssel2 3563	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ ℤ)
29		zre 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℝ)
30		zre 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 ∈ ℤ → 𝑡 ∈ ℝ)
31		letri3 10002	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑚 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚)))
32	29, 30, 31	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚)))
33		zleltp1 11305	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ 𝑡 < (𝑚 + 1)))
34		peano2re 10088	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ ℝ → (𝑚 + 1) ∈ ℝ)
35	29, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ ℤ → (𝑚 + 1) ∈ ℝ)
36		ltnle 9996	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑡 ∈ ℝ ∧ (𝑚 + 1) ∈ ℝ) → (𝑡 < (𝑚 + 1) ↔ ¬ (𝑚 + 1) ≤ 𝑡))
37	30, 35, 36	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 < (𝑚 + 1) ↔ ¬ (𝑚 + 1) ≤ 𝑡))
38	33, 37	bitrd 267	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ ¬ (𝑚 + 1) ≤ 𝑡))
39	38	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ ¬ (𝑚 + 1) ≤ 𝑡))
40	39	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → ((𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚) ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡)))
41	32, 40	bitrd 267	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡)))
42	28, 41	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡)))
43		eleq1a 2683	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ 𝑆 → (𝑚 = 𝑡 → 𝑚 ∈ 𝑆))
44	43	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 = 𝑡 → 𝑚 ∈ 𝑆))
45	42, 44	sylbird 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → ((𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡) → 𝑚 ∈ 𝑆))
46	45	expd 451	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 ≤ 𝑡 → (¬ (𝑚 + 1) ≤ 𝑡 → 𝑚 ∈ 𝑆)))
47		con1 142	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((¬ (𝑚 + 1) ≤ 𝑡 → 𝑚 ∈ 𝑆) → (¬ 𝑚 ∈ 𝑆 → (𝑚 + 1) ≤ 𝑡))
48	46, 47	syl6 34	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 ≤ 𝑡 → (¬ 𝑚 ∈ 𝑆 → (𝑚 + 1) ≤ 𝑡)))
49	48	com23 84	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (¬ 𝑚 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡)))
50	49	exp32 629	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℤ → (𝑆 ⊆ ℤ → (𝑡 ∈ 𝑆 → (¬ 𝑚 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡)))))
51	50	com34 89	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ ℤ → (𝑆 ⊆ ℤ → (¬ 𝑚 ∈ 𝑆 → (𝑡 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡)))))
52	51	imp41 617	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬ 𝑚 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡))
53	52	ralimdva 2945	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬ 𝑚 ∈ 𝑆) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))
54	53	ex 449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) → (¬ 𝑚 ∈ 𝑆 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)))
55	27, 54	sylan9r 688	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)))
56	55	pm2.43d 51	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))
57	56	expl 646	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℤ → ((𝑆 ⊆ ℤ ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)))
58	22, 57	syl 17	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ_≥‘𝑀) → ((𝑆 ⊆ ℤ ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)))
59	21, 58	sylani 684	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ_≥‘𝑀) → ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)))
60	59	a2d 29	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ_≥‘𝑀) → (((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡) → ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)))
61	3, 6, 9, 12, 18, 60	uzind4 11622	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡))
62		breq1 4586	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → (𝑗 ≤ 𝑡 ↔ 𝑛 ≤ 𝑡))
63	62	ralbidv 2969	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑛 → (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡))
64	63	rspcev 3282	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)
65	64	expcom 450	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡 → (𝑛 ∈ 𝑆 → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡))
66	65	con3rr3 150	. . . . . . . . . 10 ⊢ (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆))
67	66	adantl 481	. . . . . . . . 9 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆))
68	61, 67	sylcom 30	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆))
69		ssel 3562	. . . . . . . . . 10 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (𝑛 ∈ 𝑆 → 𝑛 ∈ (ℤ_≥‘𝑀)))
70	69	con3rr3 150	. . . . . . . . 9 ⊢ (¬ 𝑛 ∈ (ℤ_≥‘𝑀) → (𝑆 ⊆ (ℤ_≥‘𝑀) → ¬ 𝑛 ∈ 𝑆))
71	70	adantrd 483	. . . . . . . 8 ⊢ (¬ 𝑛 ∈ (ℤ_≥‘𝑀) → ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆))
72	68, 71	pm2.61i 175	. . . . . . 7 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆)
73	72	ex 449	. . . . . 6 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆))
74	73	alrimdv 1844	. . . . 5 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → ∀𝑛 ¬ 𝑛 ∈ 𝑆))
75		eq0 3888	. . . . 5 ⊢ (𝑆 = ∅ ↔ ∀𝑛 ¬ 𝑛 ∈ 𝑆)
76	74, 75	syl6ibr 241	. . . 4 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → 𝑆 = ∅))
77	76	necon1ad 2799	. . 3 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (𝑆 ≠ ∅ → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡))
78	77	imp 444	. 2 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)
79		breq2 4587	. . . 4 ⊢ (𝑡 = 𝑘 → (𝑗 ≤ 𝑡 ↔ 𝑗 ≤ 𝑘))
80	79	cbvralv 3147	. . 3 ⊢ (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)
81	80	rexbii 3023	. 2 ⊢ (∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)
82	78, 81	sylib 207	1 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 ℤcz 11254 ℤ_≥cuz 11563

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564

This theorem is referenced by: uzwo2 11628 nnwo 11629 infssuzle 11647 infssuzcl 11648 uzwo4 38246

W3C validator