Theorem supinf 42254

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 9337	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, < ) ∈ 𝐴)
4		breq1 5092	. . . . . 6 ⊢ (𝑥 = sup(𝐵, 𝐴, < ) → (𝑥 < 𝑤 ↔ sup(𝐵, 𝐴, < ) < 𝑤))
5	4	notbid 318	. . . . 5 ⊢ (𝑥 = sup(𝐵, 𝐴, < ) → (¬ 𝑥 < 𝑤 ↔ ¬ sup(𝐵, 𝐴, < ) < 𝑤))
6	5	ralbidv 3153	. . . 4 ⊢ (𝑥 = sup(𝐵, 𝐴, < ) → (∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, < ) < 𝑤))
7	1, 2	supub 9338	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, < ) < 𝑣))
8	7	ralrimiv 3121	. . . . 5 ⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ¬ sup(𝐵, 𝐴, < ) < 𝑣)
9		breq2 5093	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (sup(𝐵, 𝐴, < ) < 𝑣 ↔ sup(𝐵, 𝐴, < ) < 𝑤))
10	9	notbid 318	. . . . . 6 ⊢ (𝑣 = 𝑤 → (¬ sup(𝐵, 𝐴, < ) < 𝑣 ↔ ¬ sup(𝐵, 𝐴, < ) < 𝑤))
11	10	cbvralvw 3208	. . . . 5 ⊢ (∀𝑣 ∈ 𝐵 ¬ sup(𝐵, 𝐴, < ) < 𝑣 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, < ) < 𝑤)
12	8, 11	sylib 218	. . . 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 5092	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → (𝑥 < 𝑤 ↔ 𝑣 < 𝑤))
15	14	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (¬ 𝑥 < 𝑤 ↔ ¬ 𝑣 < 𝑤))
16	15	ralbidv 3153	. . . . . 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 5093	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑦 < 𝑧 ↔ 𝑦 < 𝑤))
19	18	cbvrexvw 3209	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ 𝐵 𝑦 < 𝑧 ↔ ∃𝑤 ∈ 𝐵 𝑦 < 𝑤)
20	19	imbi2i 336	. . . . . . . . . 10 ⊢ ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐵 𝑦 < 𝑤))
21	20	ralbii 3076	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐵 𝑦 < 𝑤))
22	21	anbi2i 623	. . . . . . . 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 3077	. . . . . . 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 218	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑤 ∈ 𝐵 𝑦 < 𝑤)))
25	1, 24	supnub 9341	. . . . 5 ⊢ (𝜑 → ((𝑣 ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑣 < 𝑤) → ¬ 𝑣 < sup(𝐵, 𝐴, < )))
26	17, 25	biimtrid 242	. . . 4 ⊢ (𝜑 → (𝑣 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤} → ¬ 𝑣 < sup(𝐵, 𝐴, < )))
27	26	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}) → ¬ 𝑣 < sup(𝐵, 𝐴, < ))
28	1, 3, 13, 27	infmin 9375	. 2 ⊢ (𝜑 → inf({𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}, 𝐴, < ) = sup(𝐵, 𝐴, < ))
29	28	eqcomd 2736	1 ⊢ (𝜑 → sup(𝐵, 𝐴, < ) = inf({𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}, 𝐴, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045  ∃wrex 3054  {crab 3393   class class class wbr 5089   Or wor 5521  supcsup 9319  infcinf 9320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-po 5522  df-so 5523  df-cnv 5622  df-iota 6433  df-riota 7298  df-sup 9321  df-inf 9322

This theorem is referenced by: (None)