Theorem mo2v 2289

Description: Alternate definition of "at most one." Unlike mo2 2293, which is slightly more general, it does not depend on ax-11 1842 and ax-13 1999, whence it is preferable within predicate logic. Elsewhere, most theorems depend on these axioms anyway, so this advantage is no longer important. (Contributed by Wolf Lammen, 27-May-2019.) (Proof shortened by Wolf Lammen, 10-Nov-2019.)

Assertion

Ref	Expression
mo2v

Distinct variable groups:   ,   ,

Proof of Theorem mo2v

Step	Hyp	Ref	Expression
1		df-mo 2287	. 2
2		df-eu 2286	. . 3
3	2	imbi2i 312	. 2
4		alnex 1614	. . . . . . 7
5		pm2.21 108	. . . . . . . 8
6	5	alimi 1633	. . . . . . 7
7	4, 6	sylbir 213	. . . . . 6
8	7	eximi 1656	. . . . 5
9	8	19.23bi 1871	. . . 4
10		bi1 186	. . . . . 6
11	10	alimi 1633	. . . . 5
12	11	eximi 1656	. . . 4
13	9, 12	ja 161	. . 3
14		nfia1 1954	. . . . . 6
15		id 22	. . . . . . . . . 10
16		ax12v 1855	. . . . . . . . . . 11
17	16	com12 31	. . . . . . . . . 10
18	15, 17	embantd 54	. . . . . . . . 9
19	18	spsd 1867	. . . . . . . 8
20	19	ancld 553	. . . . . . 7
21		albiim 1699	. . . . . . 7
22	20, 21	syl6ibr 227	. . . . . 6
23	14, 22	exlimi 1912	. . . . 5
24	23	eximdv 1710	. . . 4
25	24	com12 31	. . 3
26	13, 25	impbii 188	. 2
27	1, 3, 26	3bitri 271	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  E!weu 2282  E*wmo 2283

This theorem is referenced by:  mo2  2293  eu3v  2312  mo3  2323  mo3OLD  2324  sbmo  2335  moim  2339  mopick  2356  2mo2  2372  mo2icl  3278  moabex  4711  dffun3  5604  dffun6f  5607  grothprim  9233  wl-mo2dnae  30019  wl-mo2t  30020  wl-mo3t  30021

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-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