Theorem supub 9343

Description: A supremum is an upper bound. See also supcl 9342 and suplub 9344.

This proof demonstrates how to expand an iota-based definition (df-iota 6437) using riotacl2 7319.

(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 4027	. . . 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 9339	. . . . 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 9338	. . . . . 6 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
8		riotacl2 7319	. . . . . 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 2831	. . . 4 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
11	3, 10	sselid 3932	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦})
12		breq2 5095	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑤))
13	12	notbid 318	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑤))
14	13	cbvralvw 3210	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤 ∈ 𝐵 ¬ 𝑥𝑅𝑤)
15		breq1 5094	. . . . . . . 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 3155	. . . . . 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 3647	. . . 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 5095	. . . 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 3574	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  ∃!wreu 3344  {crab 3395   class class class wbr 5091   Or wor 5523  ℩crio 7302  supcsup 9324

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-po 5524  df-so 5525  df-iota 6437  df-riota 7303  df-sup 9326

This theorem is referenced by:  suplub2  9345  supssd  9347  supgtoreq  9355  supiso  9360  inflb  9374  suprub  12080  suprzub  12834  supxrun  13212  supxrub  13220  dgrub  26164  ssnnssfz  32765  oddpwdc  34362  itg2addnclem  37710  supubt  37778  supinf  42274  sn-suprubd  42526  ssnn0ssfz  48379