Theorem supub 9410

Description: A supremum is an upper bound. See also supcl 9409 and suplub 9411.

This proof demonstrates how to expand an iota-based definition (df-iota 6464) using riotacl2 7360.

(Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypotheses

Ref	Expression
supmo.1	⊢ (𝜑 → 𝑅 Or 𝐴)
supcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧

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

Proof of Theorem supub

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦)
2	1	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦))
3	2	ss2rabi 4040	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ⊆ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦}
4		supmo.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
5	4	supval2 9406	. . . . 5 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
6		supcl.2	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
7	4, 6	supeu 9405	. . . . . 6 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
8		riotacl2 7360	. . . . . 6 ⊢ (∃!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 17	. . . . 5 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
10	5, 9	eqeltrd 2828	. . . 4 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
11	3, 10	sselid 3944	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦})
12		breq2 5111	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑤))
13	12	notbid 318	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑤))
14	13	cbvralvw 3215	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤 ∈ 𝐵 ¬ 𝑥𝑅𝑤)
15		breq1 5110	. . . . . . . 8 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑥𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
16	15	notbid 318	. . . . . . 7 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (¬ 𝑥𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
17	16	ralbidv 3156	. . . . . 6 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤 ∈ 𝐵 ¬ 𝑥𝑅𝑤 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
18	14, 17	bitrid 283	. . . . 5 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
19	18	elrab 3659	. . . 4 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
20	19	simprbi 496	. . 3 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦} → ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
21	11, 20	syl 17	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
22		breq2 5111	. . . 4 ⊢ (𝑤 = 𝐶 → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
23	22	notbid 318	. . 3 ⊢ (𝑤 = 𝐶 → (¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
24	23	rspccv 3585	. 2 ⊢ (∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
25	21, 24	syl 17	1 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ∃!wreu 3352  {crab 3405   class class class wbr 5107   Or wor 5545  ℩crio 7343  supcsup 9391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-po 5546  df-so 5547  df-iota 6464  df-riota 7344  df-sup 9393

This theorem is referenced by:  suplub2  9412  supssd  9414  supgtoreq  9422  supiso  9427  inflb  9441  suprub  12144  suprzub  12898  supxrun  13276  supxrub  13284  dgrub  26139  ssnnssfz  32710  oddpwdc  34345  itg2addnclem  37665  supubt  37733  supinf  42230  sn-suprubd  42482  ssnn0ssfz  48337