Theorem supubti 7058

Description: A supremum is an upper bound. See also supclti 7057 and suplubti 7059.

This proof demonstrates how to expand an iota-based definition (df-iota 5215) using riotacl2 5887.

(Contributed by Jim Kingdon, 24-Nov-2021.)

Hypotheses

Ref	Expression
supmoti.ti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supclti.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

Assertion

Ref	Expression
supubti	⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))

Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑦,𝐴,𝑥,𝑧   𝑥,𝐵,𝑦,𝑧   𝑢,𝑅,𝑣,𝑥   𝑦,𝑅,𝑧   𝜑,𝑢,𝑣,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑣,𝑢)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑢)

Proof of Theorem supubti

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦)
2	1	a1i 9	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦))
3	2	ss2rabi 3261	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ⊆ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦}
4		supmoti.ti	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
5		supclti.2	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
6	4, 5	supval2ti 7054	. . . 4 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
7	4, 5	supeuti 7053	. . . . 5 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
8		riotacl2 5887	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
9	7, 8	syl 14	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
10	6, 9	eqeltrd 2270	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
11	3, 10	sselid 3177	. 2 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦})
12		breq2 4033	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑤))
13	12	notbid 668	. . . . . 6 ⊢ (𝑦 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑤))
14	13	cbvralv 2726	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤 ∈ 𝐵 ¬ 𝑥𝑅𝑤)
15		breq1 4032	. . . . . . 7 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑥𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
16	15	notbid 668	. . . . . 6 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (¬ 𝑥𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
17	16	ralbidv 2494	. . . . 5 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤 ∈ 𝐵 ¬ 𝑥𝑅𝑤 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
18	14, 17	bitrid 192	. . . 4 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
19	18	elrab 2916	. . 3 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
20	19	simprbi 275	. 2 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦} → ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
21		breq2 4033	. . . 4 ⊢ (𝑤 = 𝐶 → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
22	21	notbid 668	. . 3 ⊢ (𝑤 = 𝐶 → (¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
23	22	rspccv 2861	. 2 ⊢ (∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
24	11, 20, 23	3syl 17	1 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ∃!wreu 2474  {crab 2476   class class class wbr 4029  ℩crio 5872  supcsup 7041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-riota 5873  df-sup 7043

This theorem is referenced by:  suplub2ti  7060  supisoti  7069  inflbti  7083  suprubex  8970  zsupcl  12084  dvdslegcd  12101