Theorem supub 8923

Description: A supremum is an upper bound. See also supcl 8922 and suplub 8924.

This proof demonstrates how to expand an iota-based definition (df-iota 6314) using riotacl2 7130.

(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 485	. . . . . 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 4053	. . . 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 8919	. . . . 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 8918	. . . . . 6 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
8		riotacl2 7130	. . . . . 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 2913	. . . 4 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
11	3, 10	sseldi 3965	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦})
12		breq2 5070	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑤))
13	12	notbid 320	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑤))
14	13	cbvralvw 3449	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤 ∈ 𝐵 ¬ 𝑥𝑅𝑤)
15		breq1 5069	. . . . . . . 8 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑥𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
16	15	notbid 320	. . . . . . 7 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (¬ 𝑥𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
17	16	ralbidv 3197	. . . . . 6 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤 ∈ 𝐵 ¬ 𝑥𝑅𝑤 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
18	14, 17	syl5bb 285	. . . . 5 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤 ∈ 𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
19	18	elrab 3680	. . . 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 499	. . 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 5070	. . . 4 ⊢ (𝑤 = 𝐶 → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
23	22	notbid 320	. . 3 ⊢ (𝑤 = 𝐶 → (¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
24	23	rspccv 3620	. 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 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  {crab 3142   class class class wbr 5066   Or wor 5473  ℩crio 7113  supcsup 8904

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-po 5474  df-so 5475  df-iota 6314  df-riota 7114  df-sup 8906

This theorem is referenced by:  suplub2  8925  supgtoreq  8934  supiso  8939  inflb  8953  suprub  11602  suprzub  12340  supxrun  12710  supxrub  12718  dgrub  24824  supssd  30445  ssnnssfz  30510  oddpwdc  31612  itg2addnclem  34958  supubt  35029  ssnn0ssfz  44417