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Theorem xrub 13262

Description: By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals. (Contributed by NM, 9-Apr-2006.)

**Assertion**
Ref	Expression
xrub	⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ ∀𝑥 ∈ ℝ^ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦

Proof of Theorem xrub Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5082	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 < 𝐵 ↔ 𝑧 < 𝐵))
2		breq1 5082	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 < 𝑦 ↔ 𝑧 < 𝑦))
3	2	rexbidv 3164	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑧 < 𝑦))
4	1, 3	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)))
5	4	cbvralvw 3218	. . . 4 ⊢ (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦))
6		elxr 13065	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
7		pm2.27 42	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → ((𝑥 ∈ ℝ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)) → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))
8	7	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → (𝑥 ∈ ℝ → ((𝑥 ∈ ℝ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)) → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))))
9		pnfnlt 13077	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
10		breq1 5082	. . . . . . . . . . . . . 14 ⊢ (𝑥 = +∞ → (𝑥 < 𝐵 ↔ +∞ < 𝐵))
11	10	notbid 319	. . . . . . . . . . . . 13 ⊢ (𝑥 = +∞ → (¬ 𝑥 < 𝐵 ↔ ¬ +∞ < 𝐵))
12	9, 11	imbitrrid 247	. . . . . . . . . . . 12 ⊢ (𝑥 = +∞ → (𝐵 ∈ ℝ^* → ¬ 𝑥 < 𝐵))
13		pm2.21 123	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 < 𝐵 → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))
14	12, 13	syl6com 37	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (𝑥 = +∞ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))
15	14	ad2antlr 733	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → (𝑥 = +∞ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))
16	15	a1dd 50	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → (𝑥 = +∞ → ((𝑥 ∈ ℝ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)) → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))))
17		elxr 13065	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
18		peano2rem 11459	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) ∈ ℝ)
19		breq1 5082	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝐵 − 1) → (𝑧 < 𝐵 ↔ (𝐵 − 1) < 𝐵))
20		breq1 5082	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (𝐵 − 1) → (𝑧 < 𝑦 ↔ (𝐵 − 1) < 𝑦))
21	20	rexbidv 3164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝐵 − 1) → (∃𝑦 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
22	19, 21	imbi12d 345	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝐵 − 1) → ((𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) ↔ ((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)))
23	22	rspcv 3563	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 − 1) ∈ ℝ → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → ((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)))
24	18, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℝ → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → ((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)))
25	24	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → ((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)))
26		ltm1 11995	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) < 𝐵)
27		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦) → ((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
28	26, 27	syl5com 31	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℝ → (((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
29	28	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) → (((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
30	18	ad2antlr 733	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝐵 − 1) ∈ ℝ)
31		mnflt 13072	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 − 1) ∈ ℝ → -∞ < (𝐵 − 1))
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐵 − 1))
33		mnfxr 11200	. . . . . . . . . . . . . . . . . . 19 ⊢ -∞ ∈ ℝ^*
34	30	rexrd 11193	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝐵 − 1) ∈ ℝ^*)
35		ssel2 3917	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
36	35	adantlr 721	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
37		xrlttr 13089	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-∞ ∈ ℝ^* ∧ (𝐵 − 1) ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((-∞ < (𝐵 − 1) ∧ (𝐵 − 1) < 𝑦) → -∞ < 𝑦))
38	33, 34, 36, 37	mp3an2i 1474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((-∞ < (𝐵 − 1) ∧ (𝐵 − 1) < 𝑦) → -∞ < 𝑦))
39	32, 38	mpand 701	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((𝐵 − 1) < 𝑦 → -∞ < 𝑦))
40	39	reximdva 3153	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) → (∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
41	25, 29, 40	3syld 60	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
42	41	a1dd 50	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
43		1re 11142	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
44		breq1 5082	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 1 → (𝑧 < 𝐵 ↔ 1 < 𝐵))
45		breq1 5082	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 1 → (𝑧 < 𝑦 ↔ 1 < 𝑦))
46	45	rexbidv 3164	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 1 → (∃𝑦 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 1 < 𝑦))
47	44, 46	imbi12d 345	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 1 → ((𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) ↔ (1 < 𝐵 → ∃𝑦 ∈ 𝐴 1 < 𝑦)))
48	47	rspcv 3563	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℝ → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (1 < 𝐵 → ∃𝑦 ∈ 𝐴 1 < 𝑦)))
49	43, 48	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (1 < 𝐵 → ∃𝑦 ∈ 𝐴 1 < 𝑦))
50		ltpnf 13069	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ → 1 < +∞)
51	43, 50	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < +∞
52		breq2 5083	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = +∞ → (1 < 𝐵 ↔ 1 < +∞))
53	51, 52	mpbiri 259	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = +∞ → 1 < 𝐵)
54		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 < 𝐵 → ∃𝑦 ∈ 𝐴 1 < 𝑦) → (1 < 𝐵 → ∃𝑦 ∈ 𝐴 1 < 𝑦))
55	53, 54	syl5com 31	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = +∞ → ((1 < 𝐵 → ∃𝑦 ∈ 𝐴 1 < 𝑦) → ∃𝑦 ∈ 𝐴 1 < 𝑦))
56		mnflt 13072	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 ∈ ℝ → -∞ < 1)
57	43, 56	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ -∞ < 1
58		rexr 11189	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 ∈ ℝ → 1 ∈ ℝ^*)
59	43, 58	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ^*
60		xrlttr 13089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-∞ ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((-∞ < 1 ∧ 1 < 𝑦) → -∞ < 𝑦))
61	33, 59, 60	mp3an12 1459	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ^* → ((-∞ < 1 ∧ 1 < 𝑦) → -∞ < 𝑦))
62	57, 61	mpani 702	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ^* → (1 < 𝑦 → -∞ < 𝑦))
63	35, 62	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → (1 < 𝑦 → -∞ < 𝑦))
64	63	reximdva 3153	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ⊆ ℝ^* → (∃𝑦 ∈ 𝐴 1 < 𝑦 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
65	55, 64	sylan9r 513	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 = +∞) → ((1 < 𝐵 → ∃𝑦 ∈ 𝐴 1 < 𝑦) → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
66	49, 65	syl5 34	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 = +∞) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
67	66	a1dd 50	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 = +∞) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
68		xrltnr 13068	. . . . . . . . . . . . . . . . . . 19 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
69	33, 68	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ -∞ < -∞
70		breq2 5083	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 ↔ -∞ < -∞))
71	69, 70	mtbiri 328	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = -∞ → ¬ -∞ < 𝐵)
72	71	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 = -∞) → ¬ -∞ < 𝐵)
73	72	pm2.21d 121	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 = -∞) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
74	73	a1d 25	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 = -∞) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
75	42, 67, 74	3jaodan 1439	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ^* ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
76	17, 75	sylan2b 600	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
77	76	imp 407	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
78		breq1 5082	. . . . . . . . . . . 12 ⊢ (𝑥 = -∞ → (𝑥 < 𝐵 ↔ -∞ < 𝐵))
79		breq1 5082	. . . . . . . . . . . . 13 ⊢ (𝑥 = -∞ → (𝑥 < 𝑦 ↔ -∞ < 𝑦))
80	79	rexbidv 3164	. . . . . . . . . . . 12 ⊢ (𝑥 = -∞ → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
81	78, 80	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑥 = -∞ → ((𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
82	77, 81	syl5ibrcom 248	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → (𝑥 = -∞ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))
83	82	a1dd 50	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → (𝑥 = -∞ → ((𝑥 ∈ ℝ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)) → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))))
84	8, 16, 83	3jaod 1437	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → ((𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞) → ((𝑥 ∈ ℝ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)) → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))))
85	6, 84	biimtrid 243	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → (𝑥 ∈ ℝ^ → ((𝑥 ∈ ℝ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)) → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))))
86	85	com23 86	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → ((𝑥 ∈ ℝ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)) → (𝑥 ∈ ℝ^ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))))
87	86	ralimdv2 3149	. . . . 5 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) → ∀𝑥 ∈ ℝ^ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))
88	87	ex 413	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) → ∀𝑥 ∈ ℝ^ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))))
89	5, 88	biimtrid 243	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) → (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) → ∀𝑥 ∈ ℝ^ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))))
90	89	pm2.43d 53	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) → ∀𝑥 ∈ ℝ^ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))
91		rexr 11189	. . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
92	91	imim1i 63	. . 3 ⊢ ((𝑥 ∈ ℝ^* → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)) → (𝑥 ∈ ℝ → (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))
93	92	ralimi2 3072	. 2 ⊢ (∀𝑥 ∈ ℝ^* (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) → ∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))
94	90, 93	impbid1 226	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ ∀𝑥 ∈ ℝ^ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ w3o 1091 = wceq 1547 ∈ wcel 2119 ∀wral 3054 ∃wrex 3064 ⊆ wss 3890 class class class wbr 5079 (class class class)co 7363 ℝcr 11035 1c1 11037 +∞cpnf 11174 -∞cmnf 11175 ℝ^*cxr 11176 < clt 11177 − cmin 11375

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378

This theorem is referenced by: supxr 13263

W3C validator