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Theorem opthneg 4692
 Description: Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)
Assertion
Ref Expression
opthneg

Proof of Theorem opthneg
StepHypRef Expression
1 df-ne 2618 . 2
2 opthg 4688 . . . 4
32notbid 295 . . 3
4 ianor 490 . . . 4
5 df-ne 2618 . . . . . . 7
65bicomi 205 . . . . . 6
76a1i 11 . . . . 5
8 df-ne 2618 . . . . . . 7
98bicomi 205 . . . . . 6
109a1i 11 . . . . 5
117, 10orbi12d 714 . . . 4
124, 11syl5bb 260 . . 3
133, 12bitrd 256 . 2
141, 13syl5bb 260 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wo 369   wa 370   wceq 1437   wcel 1867   wne 2616  cop 3999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000 This theorem is referenced by:  opthne  4693  zlmodzxznm  39063
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