Theorem uzwo4 45077

Description: Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
uzwo4.1	⊢ Ⅎ𝑗𝜓
uzwo4.2	⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
uzwo4	⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))

Distinct variable groups:   𝑘,𝑀   𝑆,𝑗,𝑘   𝜑,𝑘

Allowed substitution hints:   𝜑(𝑗)   𝜓(𝑗,𝑘)   𝑀(𝑗)

Proof of Theorem uzwo4

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4055	. . . . . 6 ⊢ {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ 𝑆
2	1	a1i 11	. . . . 5 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ 𝑆)
3		id 22	. . . . 5 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → 𝑆 ⊆ (ℤ≥‘𝑀))
4	2, 3	sstrd 3969	. . . 4 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ (ℤ≥‘𝑀))
5	4	adantr 480	. . 3 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ (ℤ≥‘𝑀))
6		rabn0 4364	. . . . . 6 ⊢ ({𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅ ↔ ∃𝑗 ∈ 𝑆 𝜑)
7	6	bicomi 224	. . . . 5 ⊢ (∃𝑗 ∈ 𝑆 𝜑 ↔ {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅)
8	7	biimpi 216	. . . 4 ⊢ (∃𝑗 ∈ 𝑆 𝜑 → {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅)
9	8	adantl 481	. . 3 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅)
10		uzwo 12927	. . 3 ⊢ (({𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ (ℤ≥‘𝑀) ∧ {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅) → ∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘)
11	5, 9, 10	syl2anc 584	. 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → ∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘)
12	1	sseli 3954	. . . . . . . 8 ⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → 𝑖 ∈ 𝑆)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → 𝑖 ∈ 𝑆)
14	13	3adant1 1130	. . . . . 6 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → 𝑖 ∈ 𝑆)
15		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑖
16		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑆
17	15	nfsbc1 3784	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗[𝑖 / 𝑗]𝜑
18		sbceq1a 3776	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝜑 ↔ [𝑖 / 𝑗]𝜑))
19	15, 16, 17, 18	elrabf 3667	. . . . . . . . . . 11 ⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ↔ (𝑖 ∈ 𝑆 ∧ [𝑖 / 𝑗]𝜑))
20	19	biimpi 216	. . . . . . . . . 10 ⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → (𝑖 ∈ 𝑆 ∧ [𝑖 / 𝑗]𝜑))
21	20	simprd 495	. . . . . . . . 9 ⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → [𝑖 / 𝑗]𝜑)
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → [𝑖 / 𝑗]𝜑)
23	22	3adant1 1130	. . . . . . 7 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → [𝑖 / 𝑗]𝜑)
24		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑆 ⊆ (ℤ≥‘𝑀)
25		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}
26		nfra1 3266	. . . . . . . . 9 ⊢ Ⅎ𝑘∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘
27	24, 25, 26	nf3an 1901	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘)
28		simpl13 1251	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘)
29		simpl2 1193	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → 𝑘 ∈ 𝑆)
30		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → 𝜓)
31		simpll 766	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ 𝑆) ∧ 𝜓) → ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘)
32		id 22	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝜓) → (𝑘 ∈ 𝑆 ∧ 𝜓))
33		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝑘
34		uzwo4.1	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝜓
35		uzwo4.2	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜓))
36	33, 16, 34, 35	elrabf 3667	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ↔ (𝑘 ∈ 𝑆 ∧ 𝜓))
37	32, 36	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝜓) → 𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑})
38	37	adantll 714	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ 𝑆) ∧ 𝜓) → 𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑})
39		rspa 3231	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ≤ 𝑘)
40	31, 38, 39	syl2anc 584	. . . . . . . . . . 11 ⊢ (((∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ 𝑆) ∧ 𝜓) → 𝑖 ≤ 𝑘)
41	28, 29, 30, 40	syl21anc 837	. . . . . . . . . 10 ⊢ ((((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → 𝑖 ≤ 𝑘)
42	4	sselda 3958	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ∈ (ℤ≥‘𝑀))
43		eluzelz 12862	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (ℤ≥‘𝑀) → 𝑖 ∈ ℤ)
44	42, 43	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ∈ ℤ)
45	44	zred 12697	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ∈ ℝ)
46	45	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → 𝑖 ∈ ℝ)
47	46	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → 𝑖 ∈ ℝ)
48		ssel2 3953	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ (ℤ≥‘𝑀))
49		eluzelz 12862	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ)
50	48, 49	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ ℤ)
51	50	zred 12697	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ ℝ)
52	51	3ad2antl1 1186	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ ℝ)
53	52	3adant3 1132	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → 𝑘 ∈ ℝ)
54		simp3 1138	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → 𝑘 < 𝑖)
55		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → 𝑘 < 𝑖)
56		simp2 1137	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → 𝑘 ∈ ℝ)
57		simp1 1136	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → 𝑖 ∈ ℝ)
58	56, 57	ltnled 11382	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → (𝑘 < 𝑖 ↔ ¬ 𝑖 ≤ 𝑘))
59	55, 58	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → ¬ 𝑖 ≤ 𝑘)
60	47, 53, 54, 59	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → ¬ 𝑖 ≤ 𝑘)
61	60	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → ¬ 𝑖 ≤ 𝑘)
62	41, 61	pm2.65da 816	. . . . . . . . 9 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → ¬ 𝜓)
63	62	3exp 1119	. . . . . . . 8 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → (𝑘 ∈ 𝑆 → (𝑘 < 𝑖 → ¬ 𝜓)))
64	27, 63	ralrimi 3240	. . . . . . 7 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))
65	23, 64	jca 511	. . . . . 6 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → ([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓)))
66		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑘 < 𝑖
67	34	nfn 1857	. . . . . . . . . 10 ⊢ Ⅎ𝑗 ¬ 𝜓
68	66, 67	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑘 < 𝑖 → ¬ 𝜓)
69	16, 68	nfralw 3291	. . . . . . . 8 ⊢ Ⅎ𝑗∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓)
70	17, 69	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑗([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))
71		breq2 5123	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑘 < 𝑗 ↔ 𝑘 < 𝑖))
72	71	imbi1d 341	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝑘 < 𝑗 → ¬ 𝜓) ↔ (𝑘 < 𝑖 → ¬ 𝜓)))
73	72	ralbidv 3163	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → (∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓) ↔ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓)))
74	18, 73	anbi12d 632	. . . . . . 7 ⊢ (𝑗 = 𝑖 → ((𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)) ↔ ([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))))
75	70, 74	rspce 3590	. . . . . 6 ⊢ ((𝑖 ∈ 𝑆 ∧ ([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))
76	14, 65, 75	syl2anc 584	. . . . 5 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))
77	76	3exp 1119	. . . 4 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → (∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))))
78	77	rexlimdv 3139	. . 3 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → (∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓))))
79	78	adantr 480	. 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → (∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓))))
80	11, 79	mpd 15	1 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3415  [wsbc 3765   ⊆ wss 3926  ∅c0 4308   class class class wbr 5119  ‘cfv 6531  ℝcr 11128   < clt 11269   ≤ cle 11270  ℤcz 12588  ℤ_≥cuz 12852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853

This theorem is referenced by:  iundjiun  46489