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Theorem xrub 12746

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 5035	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 < 𝐵 ↔ 𝑧 < 𝐵))
2		breq1 5035	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 < 𝑦 ↔ 𝑧 < 𝑦))
3	2	rexbidv 3221	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑧 < 𝑦))
4	1, 3	imbi12d 348	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)))
5	4	cbvralvw 3361	. . . 4 ⊢ (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦))
6		elxr 12552	. . . . . . . 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 12564	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
10		breq1 5035	. . . . . . . . . . . . . 14 ⊢ (𝑥 = +∞ → (𝑥 < 𝐵 ↔ +∞ < 𝐵))
11	10	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑥 = +∞ → (¬ 𝑥 < 𝐵 ↔ ¬ +∞ < 𝐵))
12	9, 11	syl5ibr 249	. . . . . . . . . . . 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 726	. . . . . . . . . 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 12552	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
18		peano2rem 10991	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) ∈ ℝ)
19		breq1 5035	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝐵 − 1) → (𝑧 < 𝐵 ↔ (𝐵 − 1) < 𝐵))
20		breq1 5035	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (𝐵 − 1) → (𝑧 < 𝑦 ↔ (𝐵 − 1) < 𝑦))
21	20	rexbidv 3221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝐵 − 1) → (∃𝑦 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
22	19, 21	imbi12d 348	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝐵 − 1) → ((𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) ↔ ((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)))
23	22	rspcv 3536	. . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → ((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)))
26		ltm1 11520	. . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) → (((𝐵 − 1) < 𝐵 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
30	18	ad2antlr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝐵 − 1) ∈ ℝ)
31		mnflt 12559	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 − 1) ∈ ℝ → -∞ < (𝐵 − 1))
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐵 − 1))
33		mnfxr 10736	. . . . . . . . . . . . . . . . . . 19 ⊢ -∞ ∈ ℝ^*
34	30	rexrd 10729	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝐵 − 1) ∈ ℝ^*)
35		ssel2 3887	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
36	35	adantlr 714	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
37		xrlttr 12574	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-∞ ∈ ℝ^* ∧ (𝐵 − 1) ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((-∞ < (𝐵 − 1) ∧ (𝐵 − 1) < 𝑦) → -∞ < 𝑦))
38	33, 34, 36, 37	mp3an2i 1463	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((-∞ < (𝐵 − 1) ∧ (𝐵 − 1) < 𝑦) → -∞ < 𝑦))
39	32, 38	mpand 694	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((𝐵 − 1) < 𝑦 → -∞ < 𝑦))
40	39	reximdva 3198	. . . . . . . . . . . . . . . 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 10679	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
44		breq1 5035	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 1 → (𝑧 < 𝐵 ↔ 1 < 𝐵))
45		breq1 5035	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 1 → (𝑧 < 𝑦 ↔ 1 < 𝑦))
46	45	rexbidv 3221	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 1 → (∃𝑦 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 1 < 𝑦))
47	44, 46	imbi12d 348	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 1 → ((𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) ↔ (1 < 𝐵 → ∃𝑦 ∈ 𝐴 1 < 𝑦)))
48	47	rspcv 3536	. . . . . . . . . . . . . . . . 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 12556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ → 1 < +∞)
51	43, 50	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < +∞
52		breq2 5036	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = +∞ → (1 < 𝐵 ↔ 1 < +∞))
53	51, 52	mpbiri 261	. . . . . . . . . . . . . . . . . 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 12559	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 ∈ ℝ → -∞ < 1)
57	43, 56	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ -∞ < 1
58		rexr 10725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 ∈ ℝ → 1 ∈ ℝ^*)
59	43, 58	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ^*
60		xrlttr 12574	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-∞ ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((-∞ < 1 ∧ 1 < 𝑦) → -∞ < 𝑦))
61	33, 59, 60	mp3an12 1448	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ^* → ((-∞ < 1 ∧ 1 < 𝑦) → -∞ < 𝑦))
62	57, 61	mpani 695	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ^* → (1 < 𝑦 → -∞ < 𝑦))
63	35, 62	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → (1 < 𝑦 → -∞ < 𝑦))
64	63	reximdva 3198	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ⊆ ℝ^* → (∃𝑦 ∈ 𝐴 1 < 𝑦 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
65	55, 64	sylan9r 512	. . . . . . . . . . . . . . . 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 12555	. . . . . . . . . . . . . . . . . . 19 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
69	33, 68	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ -∞ < -∞
70		breq2 5036	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 ↔ -∞ < -∞))
71	69, 70	mtbiri 330	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = -∞ → ¬ -∞ < 𝐵)
72	71	adantl 485	. . . . . . . . . . . . . . . 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 1427	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ^* ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
76	17, 75	sylan2b 596	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
77	76	imp 410	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦)) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
78		breq1 5035	. . . . . . . . . . . 12 ⊢ (𝑥 = -∞ → (𝑥 < 𝐵 ↔ -∞ < 𝐵))
79		breq1 5035	. . . . . . . . . . . . 13 ⊢ (𝑥 = -∞ → (𝑥 < 𝑦 ↔ -∞ < 𝑦))
80	79	rexbidv 3221	. . . . . . . . . . . 12 ⊢ (𝑥 = -∞ → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
81	78, 80	imbi12d 348	. . . . . . . . . . 11 ⊢ (𝑥 = -∞ → ((𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
82	77, 81	syl5ibrcom 250	. . . . . . . . . 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 1425	. . . . . . . 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	syl5bi 245	. . . . . . 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 3107	. . . . 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 416	. . . 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	syl5bi 245	. . 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 10725	. . . 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 3089	. 2 ⊢ (∀𝑥 ∈ ℝ^* (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) → ∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))
94	90, 93	impbid1 228	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 209 ∧ wa 399 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 ∀wral 3070 ∃wrex 3071 ⊆ wss 3858 class class class wbr 5032 (class class class)co 7150 ℝcr 10574 1c1 10576 +∞cpnf 10710 -∞cmnf 10711 ℝ^*cxr 10712 < clt 10713 − cmin 10908

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-op 4529 df-uni 4799 df-br 5033 df-opab 5095 df-mpt 5113 df-id 5430 df-po 5443 df-so 5444 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8299 df-en 8528 df-dom 8529 df-sdom 8530 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911

This theorem is referenced by: supxr 12747

W3C validator