Theorem 2mo2 2372

Description: This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair and ". Note: this is not expressed by ). 2eu4 2380 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.)

Assertion

Ref	Expression
2mo2

Distinct variable groups:   ,,,   ,,

Proof of Theorem 2mo2

Step	Hyp	Ref	Expression
1		eeanv 1988	. 2
2		jcab 863	. . . . 5
3	2	2albii 1641	. . . 4
4		19.26-2 1681	. . . 4
5		19.23v 1760	. . . . . 6
6	5	albii 1640	. . . . 5
7		alcom 1845	. . . . . 6
8		19.23v 1760	. . . . . . 7
9	8	albii 1640	. . . . . 6
10	7, 9	bitri 249	. . . . 5
11	6, 10	anbi12i 697	. . . 4
12	3, 4, 11	3bitri 271	. . 3
13	12	2exbii 1668	. 2
14		mo2v 2289	. . 3
15		mo2v 2289	. . 3
16	14, 15	anbi12i 697	. 2
17	1, 13, 16	3bitr4ri 278	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  E*wmo 2283

This theorem is referenced by:  2mo  2373  2moOLD  2374  2eu4  2380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617  df-eu 2286  df-mo 2287