Theorem supinf 43261

Description: The supremum is the infimum of the upper bounds. (Contributed by SN, 29-Jun-2025.)

Hypotheses

Ref	Expression
supinf.1	⊢ (𝜑 → < Or 𝐴)
supinf.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)))

Assertion

Ref	Expression
supinf	⊢ (𝜑 → sup(𝐵, 𝐴, < ) = inf({𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}, 𝐴, < ))

Distinct variable groups:   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤, < ,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem supinf

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supinf.1	. . 3 ⊢ (𝜑 → < Or 𝐴)
2		supinf.2	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)))
3	1, 2	supcl 9434	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, < ) ∈ 𝐴)
4		breq1 5106	. . . . . 6 ⊢ (𝑥 = sup(𝐵, 𝐴, < ) → (𝑥 < 𝑤 ↔ sup(𝐵, 𝐴, < ) < 𝑤))
5	4	notbid 321	. . . . 5 ⊢ (𝑥 = sup(𝐵, 𝐴, < ) → (¬ 𝑥 < 𝑤 ↔ ¬ sup(𝐵, 𝐴, < ) < 𝑤))
6	5	ralbidv 3186	. . . 4 ⊢ (𝑥 = sup(𝐵, 𝐴, < ) → (∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, < ) < 𝑤))
7	1, 2	supub 9435	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, < ) < 𝑣))
8	7	ralrimiv 3154	. . . . 5 ⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ¬ sup(𝐵, 𝐴, < ) < 𝑣)
9		breq2 5107	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (sup(𝐵, 𝐴, < ) < 𝑣 ↔ sup(𝐵, 𝐴, < ) < 𝑤))
10	9	notbid 321	. . . . . 6 ⊢ (𝑣 = 𝑤 → (¬ sup(𝐵, 𝐴, < ) < 𝑣 ↔ ¬ sup(𝐵, 𝐴, < ) < 𝑤))
11	10	cbvralvw 3241	. . . . 5 ⊢ (∀𝑣 ∈ 𝐵 ¬ sup(𝐵, 𝐴, < ) < 𝑣 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, < ) < 𝑤)
12	8, 11	sylib 221	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, < ) < 𝑤)
13	6, 3, 12	elrabd 3647	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, < ) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤})
14		breq1 5106	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → (𝑥 < 𝑤 ↔ 𝑣 < 𝑤))
15	14	notbid 321	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (¬ 𝑥 < 𝑤 ↔ ¬ 𝑣 < 𝑤))
16	15	ralbidv 3186	. . . . . 6 ⊢ (𝑥 = 𝑣 → (∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤 ↔ ∀𝑤 ∈ 𝐵 ¬ 𝑣 < 𝑤))
17	16	elrab 3645	. . . . 5 ⊢ (𝑣 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤} ↔ (𝑣 ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑣 < 𝑤))
18		breq2 5107	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑦 < 𝑧 ↔ 𝑦 < 𝑤))
19	18	cbvrexvw 3242	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ 𝐵 𝑦 < 𝑧 ↔ ∃𝑤 ∈ 𝐵 𝑦 < 𝑤)
20	19	imbi2i 339	. . . . . . . . . 10 ⊢ ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐵 𝑦 < 𝑤))
21	20	ralbii 3109	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐵 𝑦 < 𝑤))
22	21	anbi2i 635	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐵 𝑦 < 𝑤)))
23	22	rexbii 3110	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)) ↔ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐵 𝑦 < 𝑤)))
24	2, 23	sylib 221	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐵 𝑦 < 𝑤)))
25	1, 24	supnub 9438	. . . . 5 ⊢ (𝜑 → ((𝑣 ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑣 < 𝑤) → ¬ 𝑣 < sup(𝐵, 𝐴, < )))
26	17, 25	biimtrid 245	. . . 4 ⊢ (𝜑 → (𝑣 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤} → ¬ 𝑣 < sup(𝐵, 𝐴, < )))
27	26	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}) → ¬ 𝑣 < sup(𝐵, 𝐴, < ))
28	1, 3, 13, 27	infmin 9472	. 2 ⊢ (𝜑 → inf({𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}, 𝐴, < ) = sup(𝐵, 𝐴, < ))
29	28	eqcomd 2767	1 ⊢ (𝜑 → sup(𝐵, 𝐴, < ) = inf({𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}, 𝐴, < ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413   class class class wbr 5103   Or wor 5558  supcsup 9416  infcinf 9417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-po 5559  df-so 5560  df-cnv 5659  df-iota 6487  df-riota 7369  df-sup 9418  df-inf 9419

This theorem is used by: (None)