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Theorem xrinfmsslem 12755

Description: Lemma for xrinfmss 12757. (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 3323	. . . . . 6 ⊢ (𝐴 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥))
2		rexeq 3324	. . . . . . . 8 ⊢ (𝐴 = ∅ → (∃𝑧 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑧 ∈ ∅ 𝑧 < 𝑦))
3	2	imbi2d 344	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))
4	3	ralbidv 3126	. . . . . 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 3221	. . . 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 11649	. . . . . . . 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 10738	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
9	8	anim1i 617	. . . . . . . . 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 3172	. . . . . . . 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 12565	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
13		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
14		ssel 3887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 ∈ ℝ))
15		ltpnf 12569	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℝ → 𝑧 < +∞)
16	14, 15	syl6 35	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 < +∞))
17	16	ancld 554	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ 𝑧 < +∞)))
18	17	eximdv 1918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < +∞)))
19		n0 4247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
20		df-rex 3076	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑧 ∈ 𝐴 𝑧 < +∞ ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < +∞))
21	18, 19, 20	3imtr4g 299	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ → ∃𝑧 ∈ 𝐴 𝑧 < +∞))
22	21	imp 410	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 𝑧 < +∞)
23	22	a1d 25	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞))
24	23	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ 𝑦 = +∞) → (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞))
25		breq2 5040	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = +∞ → (𝑥 < 𝑦 ↔ 𝑥 < +∞))
26		breq2 5040	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = +∞ → (𝑧 < 𝑦 ↔ 𝑧 < +∞))
27	26	rexbidv 3221	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = +∞ → (∃𝑧 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑧 ∈ 𝐴 𝑧 < +∞))
28	25, 27	imbi12d 348	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = +∞ → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞)))
29	28	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ 𝑦 = +∞) → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < +∞ → ∃𝑧 ∈ 𝐴 𝑧 < +∞)))
30	24, 29	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ 𝑦 = +∞) → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
31	30	ex 416	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) → (𝑦 = +∞ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
32	31	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ^*) ∧ (𝑦 ∈ ℝ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑦 = +∞ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
33		nltmnf 12578	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → ¬ 𝑥 < -∞)
34	33	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 = -∞) → ¬ 𝑥 < -∞)
35		breq2 5040	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = -∞ → (𝑥 < 𝑦 ↔ 𝑥 < -∞))
36	35	notbid 321	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
37	36	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 = -∞) → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
38	34, 37	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 = -∞) → ¬ 𝑥 < 𝑦)
39	38	pm2.21d 121	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 = -∞) → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
40	39	ex 416	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ^* → (𝑦 = -∞ → (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
41	40	ad2antlr 726	. . . . . . . . . . . . . 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 1425	. . . . . . . . . . . . 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	syl5bi 245	. . . . . . . . . . . 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 416	. . . . . . . . . . 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 3107	. . . . . . . . . 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 614	. . . . . . . . 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 3198	. . . . . . . 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 1129	. . . . . . 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 1115	. . . . 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 3163	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
52		rexnal 3165	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑥 ≤ 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
53		ssel2 3889	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
54		letric 10791	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
55	54	ancoms 462	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
56	55	ord 861	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
57	53, 56	sylan 583	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
58	57	an32s 651	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
59	58	reximdva 3198	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 ¬ 𝑥 ≤ 𝑦 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
60	52, 59	syl5bir 246	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
61	60	ralimdva 3108	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
62	61	imp 410	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
63	51, 62	sylan2br 597	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
64		breq1 5039	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 ≤ 𝑥 ↔ 𝑧 ≤ 𝑥))
65	64	cbvrexvw 3362	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥)
66	65	ralbii 3097	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥)
67	63, 66	sylib 221	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥)
68		mnfxr 10749	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
69		ssel 3887	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ))
70		rexr 10738	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
71		nltmnf 12578	. . . . . . . . . . . . 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 3112	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞)
75	74	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞)
76		peano2rem 11004	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) ∈ ℝ)
77		breq2 5040	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑦 − 1) → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ (𝑦 − 1)))
78	77	rexbidv 3221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑦 − 1) → (∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1)))
79	78	rspcva 3541	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 − 1) ∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1))
80	79	adantrr 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 − 1) ∈ ℝ ∧ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ)) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1))
81	80	ancoms 462	. . . . . . . . . . . . . . . . . . 19 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ (𝑦 − 1) ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1))
82	76, 81	sylan2 595	. . . . . . . . . . . . . . . . . 18 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1))
83		ssel2 3889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
84		ltm1 11533	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → (𝑦 − 1) < 𝑦)
85	84	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 − 1) < 𝑦)
86	76	ancri 553	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → ((𝑦 − 1) ∈ ℝ ∧ 𝑦 ∈ ℝ))
87		lelttr 10782	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℝ ∧ (𝑦 − 1) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ≤ (𝑦 − 1) ∧ (𝑦 − 1) < 𝑦) → 𝑧 < 𝑦))
88	87	3expb 1117	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℝ ∧ ((𝑦 − 1) ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑧 ≤ (𝑦 − 1) ∧ (𝑦 − 1) < 𝑦) → 𝑧 < 𝑦))
89	86, 88	sylan2 595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ≤ (𝑦 − 1) ∧ (𝑦 − 1) < 𝑦) → 𝑧 < 𝑦))
90	85, 89	mpan2d 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ≤ (𝑦 − 1) → 𝑧 < 𝑦))
91	83, 90	sylan 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (𝑧 ≤ (𝑦 − 1) → 𝑧 < 𝑦))
92	91	an32s 651	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → (𝑧 ≤ (𝑦 − 1) → 𝑧 < 𝑦))
93	92	reximdva 3198	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑦 − 1) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
94	93	adantll 713	. . . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . . 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 10694	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
100		breq2 5040	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 0 → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 0))
101	100	rexbidv 3221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 0 → (∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ 0))
102	101	rspcva 3541	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∃𝑧 ∈ 𝐴 𝑧 ≤ 0)
103	99, 102	mpan 689	. . . . . . . . . . . . . . . . . 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 3198	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 ∈ 𝐴 𝑧 ≤ 0 → ∃𝑧 ∈ 𝐴 𝑧 < +∞))
107	103, 106	mpan9 510	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 < +∞)
108	107, 27	syl5ibr 249	. . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . 14 ⊢ (𝑦 = +∞ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
111		xrltnr 12568	. . . . . . . . . . . . . . . . . 18 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
112	68, 111	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ¬ -∞ < -∞
113		breq2 5040	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = -∞ → (-∞ < 𝑦 ↔ -∞ < -∞))
114	112, 113	mtbiri 330	. . . . . . . . . . . . . . . 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 1424	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥 → (𝐴 ⊆ ℝ → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
118	12, 117	sylbi 220	. . . . . . . . . . . 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 410	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → (𝑦 ∈ ℝ^* → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
121	120	ralrimiv 3112	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
122	75, 121	jca 515	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 ≤ 𝑥) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
123		breq2 5040	. . . . . . . . . . . 12 ⊢ (𝑥 = -∞ → (𝑦 < 𝑥 ↔ 𝑦 < -∞))
124	123	notbid 321	. . . . . . . . . . 11 ⊢ (𝑥 = -∞ → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < -∞))
125	124	ralbidv 3126	. . . . . . . . . 10 ⊢ (𝑥 = -∞ → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞))
126		breq1 5039	. . . . . . . . . . . 12 ⊢ (𝑥 = -∞ → (𝑥 < 𝑦 ↔ -∞ < 𝑦))
127	126	imbi1d 345	. . . . . . . . . . 11 ⊢ (𝑥 = -∞ → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
128	127	ralbidv 3126	. . . . . . . . . 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 3543	. . . . . . . 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 590	. . . . . . 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 594	. . . . . 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 714	. . . . 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 812	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
135		pnfxr 10746	. . . . . 6 ⊢ +∞ ∈ ℝ^*
136		ral0 4408	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ ¬ 𝑦 < +∞
137		pnfnlt 12577	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ¬ +∞ < 𝑦)
138	137	pm2.21d 121	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))
139	138	rgen 3080	. . . . . . 7 ⊢ ∀𝑦 ∈ ℝ^* (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)
140	136, 139	pm3.2i 474	. . . . . 6 ⊢ (∀𝑦 ∈ ∅ ¬ 𝑦 < +∞ ∧ ∀𝑦 ∈ ℝ^* (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))
141		breq2 5040	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (𝑦 < 𝑥 ↔ 𝑦 < +∞))
142	141	notbid 321	. . . . . . . . 9 ⊢ (𝑥 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < +∞))
143	142	ralbidv 3126	. . . . . . . 8 ⊢ (𝑥 = +∞ → (∀𝑦 ∈ ∅ ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ ∅ ¬ 𝑦 < +∞))
144		breq1 5039	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
145	144	imbi1d 345	. . . . . . . . 9 ⊢ (𝑥 = +∞ → ((𝑥 < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦) ↔ (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)))
146	145	ralbidv 3126	. . . . . . . 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 3543	. . . . . 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 691	. . . . 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 3036	. . 3 ⊢ (𝐴 ⊆ ℝ → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
152	151	adantl 485	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
153		ssel 3887	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ^*))
154	153, 71	syl6 35	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝑦 ∈ 𝐴 → ¬ 𝑦 < -∞))
155	154	ralrimiv 3112	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞)
156		breq1 5039	. . . . . . 7 ⊢ (𝑧 = -∞ → (𝑧 < 𝑦 ↔ -∞ < 𝑦))
157	156	rspcev 3543	. . . . . 6 ⊢ ((-∞ ∈ 𝐴 ∧ -∞ < 𝑦) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)
158	157	ex 416	. . . . 5 ⊢ (-∞ ∈ 𝐴 → (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
159	158	ralrimivw 3114	. . . 4 ⊢ (-∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
160	155, 159	anim12i 615	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < -∞ ∧ ∀𝑦 ∈ ℝ^* (-∞ < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
161	68, 160, 130	sylancr 590	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ -∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ^* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
162	152, 161	jaodan 955	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 209 ∧ wa 399 ∨ wo 844 ∨ w3o 1083 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2951 ∀wral 3070 ∃wrex 3071 ⊆ wss 3860 ∅c0 4227 class class class wbr 5036 (class class class)co 7156 ℝcr 10587 0cc0 10588 1c1 10589 +∞cpnf 10723 -∞cmnf 10724 ℝ^*cxr 10725 < clt 10726 ≤ cle 10727 − cmin 10921

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 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666

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 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-po 5447 df-so 5448 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8305 df-en 8541 df-dom 8542 df-sdom 8543 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924

This theorem is referenced by: xrinfmss 12757

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