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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
StepHypRef Expression
1 df-mo 2287 . 2
2 df-eu 2286 . . 3
32imbi2i 312 . 2
4 alnex 1614 . . . . . . 7
5 pm2.21 108 . . . . . . . 8
65alimi 1633 . . . . . . 7
74, 6sylbir 213 . . . . . 6
87eximi 1656 . . . . 5
9819.23bi 1871 . . . 4
10 bi1 186 . . . . . 6
1110alimi 1633 . . . . 5
1211eximi 1656 . . . 4
139, 12ja 161 . . 3
14 nfia1 1954 . . . . . 6
15 id 22 . . . . . . . . . 10
16 ax12v 1855 . . . . . . . . . . 11
1716com12 31 . . . . . . . . . 10
1815, 17embantd 54 . . . . . . . . 9
1918spsd 1867 . . . . . . . 8
2019ancld 553 . . . . . . 7
21 albiim 1699 . . . . . . 7
2220, 21syl6ibr 227 . . . . . 6
2314, 22exlimi 1912 . . . . 5
2423eximdv 1710 . . . 4
2524com12 31 . . 3
2613, 25impbii 188 . 2
271, 3, 263bitri 271 1
 Colors of variables: wff setvar class 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
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