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 Description: If cardinal addition is non-empty, then both addends are non-empty. Theorem X.1.20 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.)
Assertion
Ref Expression
addcnnul ((A +c B) ≠ → (A B))

StepHypRef Expression
1 addceq1 4383 . . . 4 (A = → (A +c B) = ( +c B))
2 addccom 4406 . . . . 5 ( +c B) = (B +c )
3 addcnul1 4452 . . . . 5 (B +c ) =
42, 3eqtri 2373 . . . 4 ( +c B) =
51, 4syl6eq 2401 . . 3 (A = → (A +c B) = )
65necon3i 2555 . 2 ((A +c B) ≠ A)
7 addceq2 4384 . . . 4 (B = → (A +c B) = (A +c ))
8 addcnul1 4452 . . . 4 (A +c ) =
97, 8syl6eq 2401 . . 3 (B = → (A +c B) = )
109necon3i 2555 . 2 ((A +c B) ≠ B)
116, 10jca 518 1 ((A +c B) ≠ → (A B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516  ∅c0 3550   +c cplc 4375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-sik 4192  df-ssetk 4193  df-addc 4378 This theorem is referenced by:  preaddccan2  4455  leltfintr  4458  ltfintri  4466  tfinltfinlem1  4500  evenoddnnnul  4514  evenodddisj  4516  oddtfin  4518
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