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Theorem uzwo 11957

Description: Well-ordering principle: any nonempty subset of an upper set of integers has a 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 4814	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑀 → (ℎ ≤ 𝑡 ↔ 𝑀 ≤ 𝑡))
2	1	ralbidv 3133	. . . . . . . . . . 11 ⊢ (ℎ = 𝑀 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡))
3	2	imbi2d 331	. . . . . . . . . 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 4814	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑚 → (ℎ ≤ 𝑡 ↔ 𝑚 ≤ 𝑡))
5	4	ralbidv 3133	. . . . . . . . . . 11 ⊢ (ℎ = 𝑚 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡))
6	5	imbi2d 331	. . . . . . . . . 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 4814	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑚 + 1) → (ℎ ≤ 𝑡 ↔ (𝑚 + 1) ≤ 𝑡))
8	7	ralbidv 3133	. . . . . . . . . . 11 ⊢ (ℎ = (𝑚 + 1) → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))
9	8	imbi2d 331	. . . . . . . . . 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 4814	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑛 → (ℎ ≤ 𝑡 ↔ 𝑛 ≤ 𝑡))
11	10	ralbidv 3133	. . . . . . . . . . 11 ⊢ (ℎ = 𝑛 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡))
12	11	imbi2d 331	. . . . . . . . . 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 3757	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (𝑡 ∈ 𝑆 → 𝑡 ∈ (ℤ_≥‘𝑀)))
14		eluzle 11904	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑡)
15	13, 14	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (𝑡 ∈ 𝑆 → 𝑀 ≤ 𝑡))
16	15	ralrimiv 3112	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡)
17	16	adantr 472	. . . . . . . . . . 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 11911	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
20		sstr 3771	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ (ℤ_≥‘𝑀) ⊆ ℤ) → 𝑆 ⊆ ℤ)
21	19, 20	mpan2 682	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → 𝑆 ⊆ ℤ)
22		eluzelz 11901	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ℤ_≥‘𝑀) → 𝑚 ∈ ℤ)
23		breq1 4814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑚 → (𝑗 ≤ 𝑡 ↔ 𝑚 ≤ 𝑡))
24	23	ralbidv 3133	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑚 → (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡))
25	24	rspcev 3462	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)
26	25	expcom 402	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → (𝑚 ∈ 𝑆 → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡))
27	26	con3rr3 152	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ¬ 𝑚 ∈ 𝑆))
28		ssel2 3758	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ ℤ)
29		zre 11632	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℝ)
30		zre 11632	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 ∈ ℤ → 𝑡 ∈ ℝ)
31		letri3 10381	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑚 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚)))
32	29, 30, 31	syl2an 589	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚)))
33		zleltp1 11680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ 𝑡 < (𝑚 + 1)))
34		peano2re 10467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ ℝ → (𝑚 + 1) ∈ ℝ)
35	29, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ ℤ → (𝑚 + 1) ∈ ℝ)
36		ltnle 10375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑡 ∈ ℝ ∧ (𝑚 + 1) ∈ ℝ) → (𝑡 < (𝑚 + 1) ↔ ¬ (𝑚 + 1) ≤ 𝑡))
37	30, 35, 36	syl2an 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 < (𝑚 + 1) ↔ ¬ (𝑚 + 1) ≤ 𝑡))
38	33, 37	bitrd 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ ¬ (𝑚 + 1) ≤ 𝑡))
39	38	ancoms 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ ¬ (𝑚 + 1) ≤ 𝑡))
40	39	anbi2d 622	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → ((𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚) ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡)))
41	32, 40	bitrd 270	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡)))
42	28, 41	sylan2 586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡)))
43		eleq1a 2839	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ 𝑆 → (𝑚 = 𝑡 → 𝑚 ∈ 𝑆))
44	43	ad2antll 720	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 = 𝑡 → 𝑚 ∈ 𝑆))
45	42, 44	sylbird 251	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → ((𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡) → 𝑚 ∈ 𝑆))
46	45	expd 404	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 ≤ 𝑡 → (¬ (𝑚 + 1) ≤ 𝑡 → 𝑚 ∈ 𝑆)))
47		con1 145	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((¬ (𝑚 + 1) ≤ 𝑡 → 𝑚 ∈ 𝑆) → (¬ 𝑚 ∈ 𝑆 → (𝑚 + 1) ≤ 𝑡))
48	46, 47	syl6 35	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 ≤ 𝑡 → (¬ 𝑚 ∈ 𝑆 → (𝑚 + 1) ≤ 𝑡)))
49	48	com23 86	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (¬ 𝑚 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡)))
50	49	exp32 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℤ → (𝑆 ⊆ ℤ → (𝑡 ∈ 𝑆 → (¬ 𝑚 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡)))))
51	50	com34 91	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ ℤ → (𝑆 ⊆ ℤ → (¬ 𝑚 ∈ 𝑆 → (𝑡 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡)))))
52	51	imp41 416	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬ 𝑚 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡))
53	52	ralimdva 3109	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬ 𝑚 ∈ 𝑆) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))
54	53	ex 401	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) → (¬ 𝑚 ∈ 𝑆 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)))
55	27, 54	sylan9r 504	. . . . . . . . . . . . . . 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 53	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))
57	56	expl 449	. . . . . . . . . . . . 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 597	. . . . . . . . . . 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 11951	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡))
62		breq1 4814	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → (𝑗 ≤ 𝑡 ↔ 𝑛 ≤ 𝑡))
63	62	ralbidv 3133	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑛 → (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡))
64	63	rspcev 3462	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)
65	64	expcom 402	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡 → (𝑛 ∈ 𝑆 → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡))
66	65	con3rr3 152	. . . . . . . . . 10 ⊢ (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆))
67	66	adantl 473	. . . . . . . . 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 3757	. . . . . . . . . 10 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (𝑛 ∈ 𝑆 → 𝑛 ∈ (ℤ_≥‘𝑀)))
70	69	con3rr3 152	. . . . . . . . 9 ⊢ (¬ 𝑛 ∈ (ℤ_≥‘𝑀) → (𝑆 ⊆ (ℤ_≥‘𝑀) → ¬ 𝑛 ∈ 𝑆))
71	70	adantrd 485	. . . . . . . 8 ⊢ (¬ 𝑛 ∈ (ℤ_≥‘𝑀) → ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆))
72	68, 71	pm2.61i 176	. . . . . . 7 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆)
73	72	ex 401	. . . . . 6 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆))
74	73	alrimdv 2024	. . . . 5 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → ∀𝑛 ¬ 𝑛 ∈ 𝑆))
75		eq0 4095	. . . . 5 ⊢ (𝑆 = ∅ ↔ ∀𝑛 ¬ 𝑛 ∈ 𝑆)
76	74, 75	syl6ibr 243	. . . 4 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → 𝑆 = ∅))
77	76	necon1ad 2954	. . 3 ⊢ (𝑆 ⊆ (ℤ_≥‘𝑀) → (𝑆 ≠ ∅ → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡))
78	77	imp 395	. 2 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)
79		breq2 4815	. . . 4 ⊢ (𝑡 = 𝑘 → (𝑗 ≤ 𝑡 ↔ 𝑗 ≤ 𝑘))
80	79	cbvralv 3319	. . 3 ⊢ (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)
81	80	rexbii 3188	. 2 ⊢ (∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)
82	78, 81	sylib 209	1 ⊢ ((𝑆 ⊆ (ℤ_≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 197 ∧ wa 384 ∀wal 1650 = wceq 1652 ∈ wcel 2155 ≠ wne 2937 ∀wral 3055 ∃wrex 3056 ⊆ wss 3734 ∅c0 4081 class class class wbr 4811 ‘cfv 6070 (class class class)co 6846 ℝcr 10192 1c1 10194 + caddc 10196 < clt 10332 ≤ cle 10333 ℤcz 11628 ℤ_≥cuz 11891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2070 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2352 ax-ext 2743 ax-sep 4943 ax-nul 4951 ax-pow 5003 ax-pr 5064 ax-un 7151 ax-cnex 10249 ax-resscn 10250 ax-1cn 10251 ax-icn 10252 ax-addcl 10253 ax-addrcl 10254 ax-mulcl 10255 ax-mulrcl 10256 ax-mulcom 10257 ax-addass 10258 ax-mulass 10259 ax-distr 10260 ax-i2m1 10261 ax-1ne0 10262 ax-1rid 10263 ax-rnegex 10264 ax-rrecex 10265 ax-cnre 10266 ax-pre-lttri 10267 ax-pre-lttrn 10268 ax-pre-ltadd 10269 ax-pre-mulgt0 10270

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-ex 1875 df-nf 1879 df-sb 2063 df-mo 2565 df-eu 2582 df-clab 2752 df-cleq 2758 df-clel 2761 df-nfc 2896 df-ne 2938 df-nel 3041 df-ral 3060 df-rex 3061 df-reu 3062 df-rab 3064 df-v 3352 df-sbc 3599 df-csb 3694 df-dif 3737 df-un 3739 df-in 3741 df-ss 3748 df-pss 3750 df-nul 4082 df-if 4246 df-pw 4319 df-sn 4337 df-pr 4339 df-tp 4341 df-op 4343 df-uni 4597 df-iun 4680 df-br 4812 df-opab 4874 df-mpt 4891 df-tr 4914 df-id 5187 df-eprel 5192 df-po 5200 df-so 5201 df-fr 5238 df-we 5240 df-xp 5285 df-rel 5286 df-cnv 5287 df-co 5288 df-dm 5289 df-rn 5290 df-res 5291 df-ima 5292 df-pred 5867 df-ord 5913 df-on 5914 df-lim 5915 df-suc 5916 df-iota 6033 df-fun 6072 df-fn 6073 df-f 6074 df-f1 6075 df-fo 6076 df-f1o 6077 df-fv 6078 df-riota 6807 df-ov 6849 df-oprab 6850 df-mpt2 6851 df-om 7268 df-wrecs 7614 df-recs 7676 df-rdg 7714 df-er 7951 df-en 8165 df-dom 8166 df-sdom 8167 df-pnf 10334 df-mnf 10335 df-xr 10336 df-ltxr 10337 df-le 10338 df-sub 10526 df-neg 10527 df-nn 11279 df-n0 11543 df-z 11629 df-uz 11892

This theorem is referenced by: uzwo2 11958 nnwo 11959 infssuzle 11977 infssuzcl 11978 uzwo4 39896

W3C validator