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Theorem xrinfmsslem 13260

Description: Lemma for xrinfmss 13262. (Contributed by NM, 19-Jan-2006.)

**Assertion**
Ref	Expression
xrinfmsslem	⊢ ((𝐴 ⊆ ℝ^* ∧ (𝐴 ⊆ ℝ ∨ -∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))

Distinct variable group: 𝑥,𝑦,𝑧,𝐴

**Proof of Theorem xrinfmsslem**
Step	Hyp	Ref	Expression
1		raleq 3292	. . . . . 6 ⊢ (𝐴 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥))
2		rexeq 3291	. . . . . . . 8 ⊢ (𝐴 = ∅ → (∃𝑧 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑧 ∈ ∅ 𝑧 < 𝑦))
3	2	imbi2d 340	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))
4	3	ralbidv 3160	. . . . . 6 ⊢ (𝐴 = ∅ → (∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))
5	1, 4	anbi12d 633	. . . . 5 ⊢ (𝐴 = ∅ → ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))))
6	5	rexbidv 3161	. . . 4 ⊢ (𝐴 = ∅ → (∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))))
7		infm3 12115	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
8		rexr 11191	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
9	8	anim1i 616	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑥 ∈ ℝ^* ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
10	9	reximi2 3070	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
11	7, 10	syl 17	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
12		elxr 13067	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
13		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
14		ssel 3915	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 ∈ ℝ))
15		ltpnf 13071	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℝ → 𝑧 < +∞)
16	14, 15	syl6 35	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 < +∞))
17	16	ancld 550	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ 𝑧 < +∞)))
18	17	eximdv 1919	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < +∞)))
19		n0 4293	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
20		df-rex 3062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑧 ∈ 𝐴 𝑧 < +∞ ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < +∞))
21	18, 19, 20	3imtr4g 296	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ → ∃𝑧 ∈ 𝐴 𝑧 < +∞))
22	21	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 𝑧 < +∞)
23	22	a1d 25	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞))
24	23	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ 𝑦 = +∞) → (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞))
25		breq2 5089	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = +∞ → (𝑥 < 𝑦 ↔ 𝑥 < +∞))
26		breq2 5089	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = +∞ → (𝑧 < 𝑦 ↔ 𝑧 < +∞))
27	26	rexbidv 3161	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = +∞ → (∃𝑧 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑧 ∈ 𝐴 𝑧 < +∞))
28	25, 27	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = +∞ → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞)))
29	28	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ 𝑦 = +∞) → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞)))
30	24, 29	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ 𝑦 = +∞) → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
31	30	ex 412	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) → (𝑦 = +∞ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
32	31	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 = +∞ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
33		nltmnf 13080	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → ¬ 𝑥 < -∞)
34	33	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 = -∞) → ¬ 𝑥 < -∞)
35		breq2 5089	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = -∞ → (𝑥 < 𝑦 ↔ 𝑥 < -∞))
36	35	notbid 318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
37	36	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 = -∞) → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
38	34, 37	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 = -∞) → ¬ 𝑥 < 𝑦)
39	38	pm2.21d 121	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 = -∞) → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
40	39	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ^* → (𝑦 = -∞ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
41	40	ad2antlr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 = -∞ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
42	13, 32, 41	3jaod 1432	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
43	12, 42	biimtrid 242	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^) ∧ (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 ∈ ℝ^ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
44	43	ex 412	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^) → ((𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → (𝑦 ∈ ℝ^ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
45	44	ralimdv2 3146	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^) → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ ℝ^ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
46	45	anim2d 613	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
47	46	reximdva 3150	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
48	47	3adant3 1133	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → (∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
49	11, 48	mpd 15	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
50	49	3expa 1119	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
51		ralnex 3063	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
52		rexnal 3089	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑥 ≤ 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
53		ssel2 3916	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
54		letric 11246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
55	54	ancoms 458	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
56	55	ord 865	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
57	53, 56	sylan 581	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
58	57	an32s 653	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
59	58	reximdva 3150	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 ¬ 𝑥 ≤ 𝑦 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
60	52, 59	biimtrrid 243	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
61	60	ralimdva 3149	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
62	61	imp 406	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
63	51, 62	sylan2br 596	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
64		breq1 5088	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 ≤ 𝑥 ↔ 𝑧 ≤ 𝑥))
65	64	cbvrexvw 3216	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥)
66	65	ralbii 3083	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥)
67	63, 66	sylib 218	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥)
68		mnfxr 11202	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
69		ssel 3915	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ))
70		rexr 11191	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
71		nltmnf 13080	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ¬ 𝑦 < -∞)
72	70, 71	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → ¬ 𝑦 < -∞)
73	69, 72	syl6 35	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < -∞))
74	73	ralrimiv 3128	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞)
75	74	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞)
76		peano2rem 11461	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ)
77		breq2 5089	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑦 − 1) → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ (𝑦 − 1)))
78	77	rexbidv 3161	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑦 − 1) → (∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1)))
79	78	rspcva 3562	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 − 1) ∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1))
80	79	adantrr 718	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 − 1) ∈ ℝ ∧ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ)) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1))
81	80	ancoms 458	. . . . . . . . . . . . . . . . . . 19 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ (𝑦 − 1) ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1))
82	76, 81	sylan2 594	. . . . . . . . . . . . . . . . . 18 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1))
83		ssel2 3916	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
84		ltm1 11997	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) < 𝑦)
85	84	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 − 1) < 𝑦)
86	76	ancri 549	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → ((𝑦 − 1) ∈ ℝ ∧ 𝑦 ∈ ℝ))
87		lelttr 11236	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℝ ∧ (𝑦 − 1) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ≤ (𝑦 − 1) ∧ (𝑦 − 1) < 𝑦) → 𝑧 < 𝑦))
88	87	3expb 1121	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℝ ∧ ((𝑦 − 1) ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑧 ≤ (𝑦 − 1) ∧ (𝑦 − 1) < 𝑦) → 𝑧 < 𝑦))
89	86, 88	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ≤ (𝑦 − 1) ∧ (𝑦 − 1) < 𝑦) → 𝑧 < 𝑦))
90	85, 89	mpan2d 695	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ≤ (𝑦 − 1) → 𝑧 < 𝑦))
91	83, 90	sylan 581	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (𝑧 ≤ (𝑦 − 1) → 𝑧 < 𝑦))
92	91	an32s 653	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝑧 ≤ (𝑦 − 1) → 𝑧 < 𝑦))
93	92	reximdva 3150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
94	93	adantll 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
95	82, 94	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)
96	95	exp31 419	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (𝑦 ∈ ℝ → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
97	96	a1dd 50	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → (𝑦 ∈ ℝ → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
98	97	com4r 94	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
99		0re 11146	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
100		breq2 5089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 0 → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 0))
101	100	rexbidv 3161	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 0 → (∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ 0))
102	101	rspcva 3562	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∃𝑧 ∈ 𝐴 𝑧 ≤ 0)
103	99, 102	mpan 691	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → ∃𝑧 ∈ 𝐴 𝑧 ≤ 0)
104	83, 15	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 < +∞)
105	104	a1d 25	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → (𝑧 ≤ 0 → 𝑧 < +∞))
106	105	reximdva 3150	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 ∈ 𝐴 𝑧 ≤ 0 → ∃𝑧 ∈ 𝐴 𝑧 < +∞))
107	103, 106	mpan9 506	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 < +∞)
108	107, 27	imbitrrid 246	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = +∞ → ((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
109	108	a1dd 50	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = +∞ → ((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
110	109	expd 415	. . . . . . . . . . . . . 14 ⊢ (𝑦 = +∞ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
111		xrltnr 13070	. . . . . . . . . . . . . . . . . 18 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
112	68, 111	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ¬ -∞ < -∞
113		breq2 5089	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = -∞ → (-∞ < 𝑦 ↔ -∞ < -∞))
114	112, 113	mtbiri 327	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = -∞ → ¬ -∞ < 𝑦)
115	114	pm2.21d 121	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = -∞ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
116	115	2a1d 26	. . . . . . . . . . . . . 14 ⊢ (𝑦 = -∞ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
117	98, 110, 116	3jaoi 1431	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
118	12, 117	sylbi 217	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
119	118	com13 88	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝑦 ∈ ℝ^* → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
120	119	imp 406	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → (𝑦 ∈ ℝ^* → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
121	120	ralrimiv 3128	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
122	75, 121	jca 511	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
123		breq2 5089	. . . . . . . . . . . 12 ⊢ (𝑥 = -∞ → (𝑦 < 𝑥 ↔ 𝑦 < -∞))
124	123	notbid 318	. . . . . . . . . . 11 ⊢ (𝑥 = -∞ → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < -∞))
125	124	ralbidv 3160	. . . . . . . . . 10 ⊢ (𝑥 = -∞ → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞))
126		breq1 5088	. . . . . . . . . . . 12 ⊢ (𝑥 = -∞ → (𝑥 < 𝑦 ↔ -∞ < 𝑦))
127	126	imbi1d 341	. . . . . . . . . . 11 ⊢ (𝑥 = -∞ → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
128	127	ralbidv 3160	. . . . . . . . . 10 ⊢ (𝑥 = -∞ → (∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
129	125, 128	anbi12d 633	. . . . . . . . 9 ⊢ (𝑥 = -∞ → ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
130	129	rspcev 3564	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
131	68, 122, 130	sylancr 588	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
132	67, 131	syldan 592	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
133	132	adantlr 716	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
134	50, 133	pm2.61dan 813	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
135		pnfxr 11199	. . . . . 6 ⊢ +∞ ∈ ℝ^*
136		ral0 4438	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ ¬ 𝑦 < +∞
137		pnfnlt 13079	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
138	137	pm2.21d 121	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))
139	138	rgen 3053	. . . . . . 7 ⊢ ∀𝑦 ∈ ℝ^* (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)
140	136, 139	pm3.2i 470	. . . . . 6 ⊢ (∀𝑦 ∈ ∅ ¬ 𝑦 < +∞ ∧ ∀𝑦 ∈ ℝ^* (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))
141		breq2 5089	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (𝑦 < 𝑥 ↔ 𝑦 < +∞))
142	141	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < +∞))
143	142	ralbidv 3160	. . . . . . . 8 ⊢ (𝑥 = +∞ → (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ ∅ ¬ 𝑦 < +∞))
144		breq1 5088	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
145	144	imbi1d 341	. . . . . . . . 9 ⊢ (𝑥 = +∞ → ((𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) ↔ (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))
146	145	ralbidv 3160	. . . . . . . 8 ⊢ (𝑥 = +∞ → (∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ^* (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))
147	143, 146	anbi12d 633	. . . . . . 7 ⊢ (𝑥 = +∞ → ((∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)) ↔ (∀𝑦 ∈ ∅ ¬ 𝑦 < +∞ ∧ ∀𝑦 ∈ ℝ^* (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))))
148	147	rspcev 3564	. . . . . 6 ⊢ ((+∞ ∈ ℝ^* ∧ (∀𝑦 ∈ ∅ ¬ 𝑦 < +∞ ∧ ∀𝑦 ∈ ℝ^* (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))
149	135, 140, 148	mp2an 693	. . . . 5 ⊢ ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))
150	149	a1i 11	. . . 4 ⊢ (𝐴 ⊆ ℝ → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))
151	6, 134, 150	pm2.61ne 3017	. . 3 ⊢ (𝐴 ⊆ ℝ → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
152	151	adantl 481	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
153		ssel 3915	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ^*))
154	153, 71	syl6 35	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → ¬ 𝑦 < -∞))
155	154	ralrimiv 3128	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞)
156		breq1 5088	. . . . . . 7 ⊢ (𝑧 = -∞ → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
157	156	rspcev 3564	. . . . . 6 ⊢ ((-∞ ∈ 𝐴 ∧ -∞ < 𝑦) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)
158	157	ex 412	. . . . 5 ⊢ (-∞ ∈ 𝐴 → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
159	158	ralrimivw 3133	. . . 4 ⊢ (-∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
160	155, 159	anim12i 614	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
161	68, 160, 130	sylancr 588	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
162	152, 161	jaodan 960	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ (𝐴 ⊆ ℝ ∨ -∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 ⊆ wss 3889 ∅c0 4273 class class class wbr 5085 (class class class)co 7367 ℝcr 11037 0cc0 11038 1c1 11039 +∞cpnf 11176 -∞cmnf 11177 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 − cmin 11377

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380

This theorem is referenced by: xrinfmss 13262

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