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Theorem xrmaxiflemlub 11215

Description: Lemma for xrmaxif 11218. A least upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon, 28-Apr-2023.)

**Hypotheses**
Ref	Expression
xrmaxiflemlub.a	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xrmaxiflemlub.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
xrmaxiflemlub.c	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
xrmaxiflemlub.clt	⊢ (𝜑 → 𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))

**Assertion**
Ref	Expression
xrmaxiflemlub	⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))

**Proof of Theorem xrmaxiflemlub**
Step	Hyp	Ref	Expression
1		xrmaxiflemlub.clt	. . 3 ⊢ (𝜑 → 𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
2		xrmaxiflemlub.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
3		xrmaxiflemlub.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		xrmaxiflemlub.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
5		xrmaxiflemcl 11212	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ^)
6	3, 4, 5	syl2anc 409	. . . 4 ⊢ (𝜑 → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ^*)
7		xrltso 9757	. . . . 5 ⊢ < Or ℝ^*
8		sowlin 4306	. . . . 5 ⊢ (( < Or ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*)) → (𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) → (𝐶 < 𝐴 ∨ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))))
9	7, 8	mpan 422	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) → (𝐶 < 𝐴 ∨ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))))
10	2, 6, 3, 9	syl3anc 1234	. . 3 ⊢ (𝜑 → (𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) → (𝐶 < 𝐴 ∨ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))))
11	1, 10	mpd 13	. 2 ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))))
12	1	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
13	3	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐴 ∈ ℝ^*)
14	4	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐵 ∈ ℝ^*)
15		simplr 526	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
16		simpr 109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → 𝐵 = +∞)
17	16	iftrued 3534	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = +∞)
18	15, 17	breqtrd 4016	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → 𝐴 < +∞)
19	18, 16	breqtrrd 4018	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
20		simplr 526	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
21		simpr 109	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → ¬ 𝐵 = +∞)
22	21	iffalsed 3537	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
23	20, 22	breqtrd 4016	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → 𝐴 < if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
24	23	adantr 274	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 < if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
25		simpr 109	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐵 = -∞)
26	25	iftrued 3534	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = 𝐴)
27	24, 26	breqtrd 4016	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 < 𝐴)
28		xrltnr 9740	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < 𝐴)
29	3, 28	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝐴 < 𝐴)
30	29	ad3antrrr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐴)
31	27, 30	pm2.21dd 616	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → 𝐴 < 𝐵)
32	23	adantr 274	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 < if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))
33		simpr 109	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = -∞)
34	33	iffalsed 3537	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
35	32, 34	breqtrd 4016	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 < if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
36	35	adantr 274	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 < if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
37		simpr 109	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 = +∞)
38	37	iftrued 3534	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = +∞)
39	36, 38	breqtrd 4016	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 < +∞)
40		nltpnft 9775	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
41	3, 40	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
42	41	ad4antr 492	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
43	37, 42	mpbid 146	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → ¬ 𝐴 < +∞)
44	39, 43	pm2.21dd 616	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ 𝐴 = +∞) → 𝐴 < 𝐵)
45	35	adantr 274	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 < if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))
46		simpr 109	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → ¬ 𝐴 = +∞)
47	46	iffalsed 3537	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))) = if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
48	45, 47	breqtrd 4016	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 < if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
49	48	adantr 274	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → 𝐴 < if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
50		simpr 109	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → 𝐴 = -∞)
51	50	iftrued 3534	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = 𝐵)
52	49, 51	breqtrd 4016	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
53	29	ad5antr 494	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 < 𝐴)
54		simp-5l 539	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝜑)
55		simp-4r 538	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = +∞)
56	54, 55	jca 304	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝜑 ∧ ¬ 𝐵 = +∞))
57		simpllr 530	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = -∞)
58	56, 57	jca 304	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞))
59		simplr 526	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = +∞)
60	58, 59	jca 304	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞))
61		simpr 109	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = -∞)
62		simplr 526	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = +∞)
63		simpr 109	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐴 = -∞)
64		elxr 9737	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
65	3, 64	sylib 121	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
66	65	ad4antr 492	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
67	62, 63, 66	ecase23d 1346	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ∈ ℝ)
68	60, 61, 67	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 ∈ ℝ)
69		simp-4r 538	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = +∞)
70		simpllr 530	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → ¬ 𝐵 = -∞)
71		elxr 9737	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
72	4, 71	sylib 121	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
73	72	ad4antr 492	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
74	69, 70, 73	ecase23d 1346	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐵 ∈ ℝ)
75	60, 61, 74	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐵 ∈ ℝ)
76	48	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 < if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))
77	61	iffalsed 3537	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )) = sup({𝐴, 𝐵}, ℝ, < ))
78	76, 77	breqtrd 4016	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 < sup({𝐴, 𝐵}, ℝ, < ))
79		maxleastlt 11183	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ∈ ℝ ∧ 𝐴 < sup({𝐴, 𝐵}, ℝ, < ))) → (𝐴 < 𝐴 ∨ 𝐴 < 𝐵))
80	68, 75, 68, 78, 79	syl22anc 1235	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐴 < 𝐴 ∨ 𝐴 < 𝐵))
81	80	orcomd 725	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → (𝐴 < 𝐵 ∨ 𝐴 < 𝐴))
82	53, 81	ecased 1345	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → 𝐴 < 𝐵)
83		xrmnfdc 9804	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = -∞)
84		exmiddc 832	. . . . . . . . . . . 12 ⊢ (DECID 𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
85	3, 83, 84	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
86	85	ad4antr 492	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
87	52, 82, 86	mpjaodan 794	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 < 𝐵)
88		xrpnfdc 9803	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = +∞)
89		exmiddc 832	. . . . . . . . . . 11 ⊢ (DECID 𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
90	3, 88, 89	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
91	90	ad3antrrr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
92	44, 87, 91	mpjaodan 794	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 < 𝐵)
93		xrmnfdc 9804	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = -∞)
94		exmiddc 832	. . . . . . . . . 10 ⊢ (DECID 𝐵 = -∞ → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
95	4, 93, 94	3syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
96	95	ad2antrr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
97	31, 92, 96	mpjaodan 794	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) ∧ ¬ 𝐵 = +∞) → 𝐴 < 𝐵)
98		xrpnfdc 9803	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = +∞)
99		exmiddc 832	. . . . . . . 8 ⊢ (DECID 𝐵 = +∞ → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
100	14, 98, 99	3syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
101	19, 97, 100	mpjaodan 794	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐴 < 𝐵)
102	13, 14, 101	xrmaxiflemab 11214	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵)
103	12, 102	breqtrd 4016	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → 𝐶 < 𝐵)
104	103	ex 114	. . 3 ⊢ (𝜑 → (𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) → 𝐶 < 𝐵))
105	104	orim2d 784	. 2 ⊢ (𝜑 → ((𝐶 < 𝐴 ∨ 𝐴 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))) → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)))
106	11, 105	mpd 13	1 ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 704 DECID wdc 830 ∨ w3o 973 ∧ w3a 974 = wceq 1349 ∈ wcel 2142 ifcif 3527 {cpr 3585 class class class wbr 3990 Or wor 4281 supcsup 6963 ℝcr 7777 +∞cpnf 7955 -∞cmnf 7956 ℝ^*cxr 7957 < clt 7958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 610 ax-in2 611 ax-io 705 ax-5 1441 ax-7 1442 ax-gen 1443 ax-ie1 1487 ax-ie2 1488 ax-8 1498 ax-10 1499 ax-11 1500 ax-i12 1501 ax-bndl 1503 ax-4 1504 ax-17 1520 ax-i9 1524 ax-ial 1528 ax-i5r 1529 ax-13 2144 ax-14 2145 ax-ext 2153 ax-coll 4105 ax-sep 4108 ax-nul 4116 ax-pow 4161 ax-pr 4195 ax-un 4419 ax-setind 4522 ax-iinf 4573 ax-cnex 7869 ax-resscn 7870 ax-1cn 7871 ax-1re 7872 ax-icn 7873 ax-addcl 7874 ax-addrcl 7875 ax-mulcl 7876 ax-mulrcl 7877 ax-addcom 7878 ax-mulcom 7879 ax-addass 7880 ax-mulass 7881 ax-distr 7882 ax-i2m1 7883 ax-0lt1 7884 ax-1rid 7885 ax-0id 7886 ax-rnegex 7887 ax-precex 7888 ax-cnre 7889 ax-pre-ltirr 7890 ax-pre-ltwlin 7891 ax-pre-lttrn 7892 ax-pre-apti 7893 ax-pre-ltadd 7894 ax-pre-mulgt0 7895 ax-pre-mulext 7896 ax-arch 7897 ax-caucvg 7898

This theorem depends on definitions: df-bi 116 df-dc 831 df-3or 975 df-3an 976 df-tru 1352 df-fal 1355 df-nf 1455 df-sb 1757 df-eu 2023 df-mo 2024 df-clab 2158 df-cleq 2164 df-clel 2167 df-nfc 2302 df-ne 2342 df-nel 2437 df-ral 2454 df-rex 2455 df-reu 2456 df-rmo 2457 df-rab 2458 df-v 2733 df-sbc 2957 df-csb 3051 df-dif 3124 df-un 3126 df-in 3128 df-ss 3135 df-nul 3416 df-if 3528 df-pw 3569 df-sn 3590 df-pr 3591 df-op 3593 df-uni 3798 df-int 3833 df-iun 3876 df-br 3991 df-opab 4052 df-mpt 4053 df-tr 4089 df-id 4279 df-po 4282 df-iso 4283 df-iord 4352 df-on 4354 df-ilim 4355 df-suc 4357 df-iom 4576 df-xp 4618 df-rel 4619 df-cnv 4620 df-co 4621 df-dm 4622 df-rn 4623 df-res 4624 df-ima 4625 df-iota 5162 df-fun 5202 df-fn 5203 df-f 5204 df-f1 5205 df-fo 5206 df-f1o 5207 df-fv 5208 df-riota 5813 df-ov 5860 df-oprab 5861 df-mpo 5862 df-1st 6123 df-2nd 6124 df-recs 6288 df-frec 6374 df-sup 6965 df-pnf 7960 df-mnf 7961 df-xr 7962 df-ltxr 7963 df-le 7964 df-sub 8096 df-neg 8097 df-reap 8498 df-ap 8505 df-div 8594 df-inn 8883 df-2 8941 df-3 8942 df-4 8943 df-n0 9140 df-z 9217 df-uz 9492 df-rp 9615 df-seqfrec 10406 df-exp 10480 df-cj 10810 df-re 10811 df-im 10812 df-rsqrt 10966 df-abs 10967

This theorem is referenced by: xrmaxiflemval 11217 xrmaxleastlt 11223

W3C validator