Theorem supnub 2162

Description: An upper bound is not less than the supremum.

Hypothesis

Ref	Expression
supmo.1	⊢ R Or A

Assertion

Ref	Expression
supnub	⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ((C ∈ A ∧ ∀z ∈ B ¬ CRz) → ¬ CRsup(B, A, R)))

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

Proof of Theorem supnub

Step	Hyp	Ref	Expression
1		supmo.1	. . . . . . . 8 ⊢ R Or A
2	1	suplub 2161	. . . . . . 7 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ((C ∈ A ∧ CRsup(B, A, R)) → ∃z ∈ B CRz))
3	2	exp3a 292	. . . . . 6 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (C ∈ A → (CRsup(B, A, R) → ∃z ∈ B CRz)))
4	3	imp 277	. . . . 5 ⊢ ((∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) ∧ C ∈ A) → (CRsup(B, A, R) → ∃z ∈ B CRz))
5		dfrex2 1212	. . . . 5 ⊢ (∃z ∈ B CRz ↔ ¬ ∀z ∈ B ¬ CRz)
6	4, 5	syl6ib 185	. . . 4 ⊢ ((∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) ∧ C ∈ A) → (CRsup(B, A, R) → ¬ ∀z ∈ B ¬ CRz))
7	6	con2d 83	. . 3 ⊢ ((∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) ∧ C ∈ A) → (∀z ∈ B ¬ CRz → ¬ CRsup(B, A, R)))
8	7	exp 291	. 2 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (C ∈ A → (∀z ∈ B ¬ CRz → ¬ CRsup(B, A, R))))
9	8	imp3a 279	1 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ((C ∈ A ∧ ∀z ∈ B ¬ CRz) → ¬ CRsup(B, A, R)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   class class class wbr 2054   Or wor 2059  supcsup 2060

This theorem is referenced by:  supnubi 2167

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

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-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-po 2128  df-so 2138  df-sup 2154

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