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Theorem xrub 13046

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 5077	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 < 𝐵 ↔ 𝑧 < 𝐵))
2		breq1 5077	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 < 𝑦 ↔ 𝑧 < 𝑦))
3	2	rexbidv 3226	. . . . . 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 3383	. . . 4 ⊢ (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ ∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦))
6		elxr 12852	. . . . . . . 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 12864	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
10		breq1 5077	. . . . . . . . . . . . . 14 ⊢ (𝑥 = +∞ → (𝑥 < 𝐵 ↔ +∞ < 𝐵))
11	10	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑥 = +∞ → (¬ 𝑥 < 𝐵 ↔ ¬ +∞ < 𝐵))
12	9, 11	syl5ibr 245	. . . . . . . . . . . 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 724	. . . . . . . . . 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 12852	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
18		peano2rem 11288	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) ∈ ℝ)
19		breq1 5077	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝐵 − 1) → (𝑧 < 𝐵 ↔ (𝐵 − 1) < 𝐵))
20		breq1 5077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (𝐵 − 1) → (𝑧 < 𝑦 ↔ (𝐵 − 1) < 𝑦))
21	20	rexbidv 3226	. . . . . . . . . . . . . . . . . . . 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 3557	. . . . . . . . . . . . . . . . . 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 11817	. . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝐵 − 1) ∈ ℝ)
31		mnflt 12859	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 − 1) ∈ ℝ → -∞ < (𝐵 − 1))
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐵 − 1))
33		mnfxr 11032	. . . . . . . . . . . . . . . . . . 19 ⊢ -∞ ∈ ℝ^*
34	30	rexrd 11025	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝐵 − 1) ∈ ℝ^*)
35		ssel2 3916	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
36	35	adantlr 712	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
37		xrlttr 12874	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-∞ ∈ ℝ^* ∧ (𝐵 − 1) ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((-∞ < (𝐵 − 1) ∧ (𝐵 − 1) < 𝑦) → -∞ < 𝑦))
38	33, 34, 36, 37	mp3an2i 1465	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((-∞ < (𝐵 − 1) ∧ (𝐵 − 1) < 𝑦) → -∞ < 𝑦))
39	32, 38	mpand 692	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((𝐵 − 1) < 𝑦 → -∞ < 𝑦))
40	39	reximdva 3203	. . . . . . . . . . . . . . . 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 10975	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
44		breq1 5077	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 1 → (𝑧 < 𝐵 ↔ 1 < 𝐵))
45		breq1 5077	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 1 → (𝑧 < 𝑦 ↔ 1 < 𝑦))
46	45	rexbidv 3226	. . . . . . . . . . . . . . . . . . 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 3557	. . . . . . . . . . . . . . . . 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 12856	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ → 1 < +∞)
51	43, 50	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < +∞
52		breq2 5078	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = +∞ → (1 < 𝐵 ↔ 1 < +∞))
53	51, 52	mpbiri 257	. . . . . . . . . . . . . . . . . 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 12859	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 ∈ ℝ → -∞ < 1)
57	43, 56	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ -∞ < 1
58		rexr 11021	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 ∈ ℝ → 1 ∈ ℝ^*)
59	43, 58	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ^*
60		xrlttr 12874	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-∞ ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((-∞ < 1 ∧ 1 < 𝑦) → -∞ < 𝑦))
61	33, 59, 60	mp3an12 1450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ^* → ((-∞ < 1 ∧ 1 < 𝑦) → -∞ < 𝑦))
62	57, 61	mpani 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ^* → (1 < 𝑦 → -∞ < 𝑦))
63	35, 62	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → (1 < 𝑦 → -∞ < 𝑦))
64	63	reximdva 3203	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ⊆ ℝ^* → (∃𝑦 ∈ 𝐴 1 < 𝑦 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦))
65	55, 64	sylan9r 509	. . . . . . . . . . . . . . . 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 12855	. . . . . . . . . . . . . . . . . . 19 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
69	33, 68	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ -∞ < -∞
70		breq2 5078	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 ↔ -∞ < -∞))
71	69, 70	mtbiri 327	. . . . . . . . . . . . . . . . 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 1429	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ^* ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (∀𝑧 ∈ ℝ (𝑧 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑧 < 𝑦) → (-∞ < 𝐵 → ∃𝑦 ∈ 𝐴 -∞ < 𝑦)))
76	17, 75	sylan2b 594	. . . . . . . . . . . 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 5077	. . . . . . . . . . . 12 ⊢ (𝑥 = -∞ → (𝑥 < 𝐵 ↔ -∞ < 𝐵))
79		breq1 5077	. . . . . . . . . . . . 13 ⊢ (𝑥 = -∞ → (𝑥 < 𝑦 ↔ -∞ < 𝑦))
80	79	rexbidv 3226	. . . . . . . . . . . 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 246	. . . . . . . . . 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 1427	. . . . . . . 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 241	. . . . . . 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 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	syl5bi 241	. . 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 11021	. . . 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 3084	. 2 ⊢ (∀𝑥 ∈ ℝ^* (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) → ∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))
94	90, 93	impbid1 224	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 205 ∧ wa 396 ∨ w3o 1085 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 1c1 10872 +∞cpnf 11006 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009 − cmin 11205

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208

This theorem is referenced by: supxr 13047

W3C validator