Theorem supeu 2158

Description: A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for supremums in general).

Hypothesis

Ref	Expression
supmo.1	⊢ R Or A

Assertion

Ref	Expression
supeu	⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ∃!x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)))

Distinct variable group(s):   x,y,z,A   x,R,y,z   x,B,y,z

Proof of Theorem supeu

Step	Hyp	Ref	Expression
1		supmo.1	. . . 4 ⊢ R Or A
2	1	supmo 2156	. . 3 ⊢ ∃*x(x ∈ A ∧ (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)))
3	2	jctr 239	. 2 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) ∧ ∃*x(x ∈ A ∧ (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)))))
4		reu5 1339	. 2 ⊢ (∃!x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) ↔ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) ∧ ∃*x(x ∈ A ∧ (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)))))
5	3, 4	sylibr 175	1 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ∃!x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∃*wmo 1008   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ∃!wreu 1203   class class class wbr 2054   Or wor 2059

This theorem is referenced by:  supcl 2159  supub 2160  suplub 2161  supeui 2163

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-po 2128  df-so 2138

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