Theorem mosubop 1911

Description: "At most one" remains true inside order pair quantification.

Hypothesis

Ref	Expression
mosubop.1	|- E*xph

Assertion

Ref	Expression
mosubop	|- E*xE.yE.z(A = <.y, z>. /\ ph)

Distinct variable group(s):   x,y,z,A

Proof of Theorem mosubop

Step	Hyp	Ref	Expression
1		hbe1 709	. . . 4 |- (E.yE.z(A = <.y, z>. /\ ph) -> A.yE.yE.z(A = <.y, z>. /\ ph))
2	1	hbmo 1033	. . 3 |- (E*xE.yE.z(A = <.y, z>. /\ ph) -> A.yE*xE.yE.z(A = <.y, z>. /\ ph))
3		hbe1 709	. . . . . 6 |- (E.z(A = <.y, z>. /\ ph) -> A.zE.z(A = <.y, z>. /\ ph))
4	3	hbex 701	. . . . 5 |- (E.yE.z(A = <.y, z>. /\ ph) -> A.zE.yE.z(A = <.y, z>. /\ ph))
5	4	hbmo 1033	. . . 4 |- (E*xE.yE.z(A = <.y, z>. /\ ph) -> A.zE*xE.yE.z(A = <.y, z>. /\ ph))
6		mosubop.1	. . . . 5 |- E*xph
7		ax-17 925	. . . . . 6 |- (A = <.y, z>. -> A.x A = <.y, z>.)
8		copsexg 1902	. . . . . 6 |- (A = <.y, z>. -> (ph <-> E.yE.z(A = <.y, z>. /\ ph)))
9	7, 8	bimod 1030	. . . . 5 |- (A = <.y, z>. -> (E*xph <-> E*xE.yE.z(A = <.y, z>. /\ ph)))
10	6, 9	mpbii 168	. . . 4 |- (A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph))
11	5, 10	19.23ai 746	. . 3 |- (E.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph))
12	2, 11	19.23ai 746	. 2 |- (E.yE.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph))
13		pm3.26 256	. . . . . . 7 |- ((A = <.y, z>. /\ ph) -> A = <.y, z>.)
14	13	19.22i 723	. . . . . 6 |- (E.z(A = <.y, z>. /\ ph) -> E.z A = <.y, z>.)
15	14	19.22i 723	. . . . 5 |- (E.yE.z(A = <.y, z>. /\ ph) -> E.yE.z A = <.y, z>.)
16	15	19.23aiv 952	. . . 4 |- (E.xE.yE.z(A = <.y, z>. /\ ph) -> E.yE.z A = <.y, z>.)
17	16	con3i 90	. . 3 |- (-. E.yE.z A = <.y, z>. -> -. E.xE.yE.z(A = <.y, z>. /\ ph))
18		exmo 1042	. . . 4 |- (E.xE.yE.z(A = <.y, z>. /\ ph) \/ E*xE.yE.z(A = <.y, z>. /\ ph))
19	18	ori 200	. . 3 |- (-. E.xE.yE.z(A = <.y, z>. /\ ph) -> E*xE.yE.z(A = <.y, z>. /\ ph))
20	17, 19	syl 12	. 2 |- (-. E.yE.z A = <.y, z>. -> E*xE.yE.z(A = <.y, z>. /\ ph))
21	12, 20	pm2.61i 110	1 |- E*xE.yE.z(A = <.y, z>. /\ ph)

Syntax hints:  -. wn 1   /\ wa 196  E.wex 678  E*wmo 1008   = wceq 1091  <.cop 1810

This theorem is referenced by:  funoprab 3037  oprabex3 3046  oprabval3 3052

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  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

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