Theorem uzwo4 41335

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 4056	. . . . . 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 3977	. . . 4 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ (ℤ≥‘𝑀))
5	4	adantr 483	. . 3 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → {𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ (ℤ≥‘𝑀))
6		rabn0 4339	. . . . . 6 ⊢ ({𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅ ↔ ∃𝑗 ∈ 𝑆 𝜑)
7	6	bicomi 226	. . . . 5 ⊢ (∃𝑗 ∈ 𝑆 𝜑 ↔ {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅)
8	7	biimpi 218	. . . 4 ⊢ (∃𝑗 ∈ 𝑆 𝜑 → {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅)
9	8	adantl 484	. . 3 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅)
10		uzwo 12312	. . 3 ⊢ (({𝑗 ∈ 𝑆 ∣ 𝜑} ⊆ (ℤ≥‘𝑀) ∧ {𝑗 ∈ 𝑆 ∣ 𝜑} ≠ ∅) → ∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘)
11	5, 9, 10	syl2anc 586	. 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → ∃𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘)
12	1	sseli 3963	. . . . . . . 8 ⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → 𝑖 ∈ 𝑆)
13	12	adantr 483	. . . . . . 7 ⊢ ((𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → 𝑖 ∈ 𝑆)
14	13	3adant1 1126	. . . . . 6 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → 𝑖 ∈ 𝑆)
15		nfcv 2977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑖
16		nfcv 2977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑆
17	15	nfsbc1 3791	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗[𝑖 / 𝑗]𝜑
18		sbceq1a 3783	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝜑 ↔ [𝑖 / 𝑗]𝜑))
19	15, 16, 17, 18	elrabf 3676	. . . . . . . . . . 11 ⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ↔ (𝑖 ∈ 𝑆 ∧ [𝑖 / 𝑗]𝜑))
20	19	biimpi 218	. . . . . . . . . 10 ⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → (𝑖 ∈ 𝑆 ∧ [𝑖 / 𝑗]𝜑))
21	20	simprd 498	. . . . . . . . 9 ⊢ (𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} → [𝑖 / 𝑗]𝜑)
22	21	adantr 483	. . . . . . . 8 ⊢ ((𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → [𝑖 / 𝑗]𝜑)
23	22	3adant1 1126	. . . . . . 7 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → [𝑖 / 𝑗]𝜑)
24		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑆 ⊆ (ℤ≥‘𝑀)
25		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}
26		nfra1 3219	. . . . . . . . 9 ⊢ Ⅎ𝑘∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘
27	24, 25, 26	nf3an 1902	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘)
28		simpl13 1246	. . . . . . . . . . 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 1188	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → 𝑘 ∈ 𝑆)
30		simpr 487	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → 𝜓)
31		simpll 765	. . . . . . . . . . . 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 2977	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝑘
34		uzwo4.1	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝜓
35		uzwo4.2	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜓))
36	33, 16, 34, 35	elrabf 3676	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ↔ (𝑘 ∈ 𝑆 ∧ 𝜓))
37	32, 36	sylibr 236	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝜓) → 𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑})
38	37	adantll 712	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ 𝑆) ∧ 𝜓) → 𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑})
39		rspa 3206	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ≤ 𝑘)
40	31, 38, 39	syl2anc 586	. . . . . . . . . . 11 ⊢ (((∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘 ∧ 𝑘 ∈ 𝑆) ∧ 𝜓) → 𝑖 ≤ 𝑘)
41	28, 29, 30, 40	syl21anc 835	. . . . . . . . . 10 ⊢ ((((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) ∧ 𝜓) → 𝑖 ≤ 𝑘)
42	4	sselda 3967	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ∈ (ℤ≥‘𝑀))
43		eluzelz 12254	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (ℤ≥‘𝑀) → 𝑖 ∈ ℤ)
44	42, 43	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ∈ ℤ)
45	44	zred 12088	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}) → 𝑖 ∈ ℝ)
46	45	3adant3 1128	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → 𝑖 ∈ ℝ)
47	46	3ad2ant1 1129	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → 𝑖 ∈ ℝ)
48		ssel2 3962	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ (ℤ≥‘𝑀))
49		eluzelz 12254	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ)
50	48, 49	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ ℤ)
51	50	zred 12088	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ ℝ)
52	51	3ad2antl1 1181	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ ℝ)
53	52	3adant3 1128	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → 𝑘 ∈ ℝ)
54		simp3 1134	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → 𝑘 < 𝑖)
55		simp3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → 𝑘 < 𝑖)
56		simp2 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → 𝑘 ∈ ℝ)
57		simp1 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → 𝑖 ∈ ℝ)
58	56, 57	ltnled 10787	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → (𝑘 < 𝑖 ↔ ¬ 𝑖 ≤ 𝑘))
59	55, 58	mpbid 234	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑘 < 𝑖) → ¬ 𝑖 ≤ 𝑘)
60	47, 53, 54, 59	syl3anc 1367	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → ¬ 𝑖 ≤ 𝑘)
61	60	adantr 483	. . . . . . . . . 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 815	. . . . . . . . 9 ⊢ (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) ∧ 𝑘 ∈ 𝑆 ∧ 𝑘 < 𝑖) → ¬ 𝜓)
63	62	3exp 1115	. . . . . . . 8 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → (𝑘 ∈ 𝑆 → (𝑘 < 𝑖 → ¬ 𝜓)))
64	27, 63	ralrimi 3216	. . . . . . 7 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))
65	23, 64	jca 514	. . . . . 6 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑖 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑} ∧ ∀𝑘 ∈ {𝑗 ∈ 𝑆 ∣ 𝜑}𝑖 ≤ 𝑘) → ([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓)))
66		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑘 < 𝑖
67	34	nfn 1857	. . . . . . . . . 10 ⊢ Ⅎ𝑗 ¬ 𝜓
68	66, 67	nfim 1897	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑘 < 𝑖 → ¬ 𝜓)
69	16, 68	nfralw 3225	. . . . . . . 8 ⊢ Ⅎ𝑗∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓)
70	17, 69	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑗([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))
71		breq2 5070	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑘 < 𝑗 ↔ 𝑘 < 𝑖))
72	71	imbi1d 344	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝑘 < 𝑗 → ¬ 𝜓) ↔ (𝑘 < 𝑖 → ¬ 𝜓)))
73	72	ralbidv 3197	. . . . . . . 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 3612	. . . . . 6 ⊢ ((𝑖 ∈ 𝑆 ∧ ([𝑖 / 𝑗]𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑖 → ¬ 𝜓))) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))
76	14, 65, 75	syl2anc 586	. . . . 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 1115	. . . 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 3283	. . 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 483	. 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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  [wsbc 3772   ⊆ wss 3936  ∅c0 4291   class class class wbr 5066  ‘cfv 6355  ℝcr 10536   < clt 10675   ≤ cle 10676  ℤcz 11982  ℤ_≥cuz 12244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245

This theorem is referenced by:  iundjiun  42762