New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  addceq0 GIF version

 Description: The sum of two cardinals is zero iff both addends are zero. (Contributed by SF, 12-Mar-2015.)
Assertion
Ref Expression
addceq0 ((A NC B NC ) → ((A +c B) = 0c ↔ (A = 0c B = 0c)))

Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 ianor 474 . . . 4 (¬ (A = 0c B = 0c) ↔ (¬ A = 0c ¬ B = 0c))
2 nc0suc 6217 . . . . . . . 8 (A NC → (A = 0c p NC A = (p +c 1c)))
32ord 366 . . . . . . 7 (A NC → (¬ A = 0cp NC A = (p +c 1c)))
43adantr 451 . . . . . 6 ((A NC B NC ) → (¬ A = 0cp NC A = (p +c 1c)))
5 addc32 4416 . . . . . . . . 9 ((p +c 1c) +c B) = ((p +c B) +c 1c)
6 0cnsuc 4401 . . . . . . . . 9 ((p +c B) +c 1c) ≠ 0c
75, 6eqnetri 2533 . . . . . . . 8 ((p +c 1c) +c B) ≠ 0c
8 addceq1 4383 . . . . . . . . . 10 (A = (p +c 1c) → (A +c B) = ((p +c 1c) +c B))
98eqeq1d 2361 . . . . . . . . 9 (A = (p +c 1c) → ((A +c B) = 0c ↔ ((p +c 1c) +c B) = 0c))
109necon3bbid 2550 . . . . . . . 8 (A = (p +c 1c) → (¬ (A +c B) = 0c ↔ ((p +c 1c) +c B) ≠ 0c))
117, 10mpbiri 224 . . . . . . 7 (A = (p +c 1c) → ¬ (A +c B) = 0c)
1211rexlimivw 2734 . . . . . 6 (p NC A = (p +c 1c) → ¬ (A +c B) = 0c)
134, 12syl6 29 . . . . 5 ((A NC B NC ) → (¬ A = 0c → ¬ (A +c B) = 0c))
14 nc0suc 6217 . . . . . . . 8 (B NC → (B = 0c p NC B = (p +c 1c)))
1514ord 366 . . . . . . 7 (B NC → (¬ B = 0cp NC B = (p +c 1c)))
1615adantl 452 . . . . . 6 ((A NC B NC ) → (¬ B = 0cp NC B = (p +c 1c)))
17 addcass 4415 . . . . . . . . 9 ((A +c p) +c 1c) = (A +c (p +c 1c))
18 0cnsuc 4401 . . . . . . . . 9 ((A +c p) +c 1c) ≠ 0c
1917, 18eqnetrri 2535 . . . . . . . 8 (A +c (p +c 1c)) ≠ 0c
20 addceq2 4384 . . . . . . . . . 10 (B = (p +c 1c) → (A +c B) = (A +c (p +c 1c)))
2120eqeq1d 2361 . . . . . . . . 9 (B = (p +c 1c) → ((A +c B) = 0c ↔ (A +c (p +c 1c)) = 0c))
2221necon3bbid 2550 . . . . . . . 8 (B = (p +c 1c) → (¬ (A +c B) = 0c ↔ (A +c (p +c 1c)) ≠ 0c))
2319, 22mpbiri 224 . . . . . . 7 (B = (p +c 1c) → ¬ (A +c B) = 0c)
2423rexlimivw 2734 . . . . . 6 (p NC B = (p +c 1c) → ¬ (A +c B) = 0c)
2516, 24syl6 29 . . . . 5 ((A NC B NC ) → (¬ B = 0c → ¬ (A +c B) = 0c))
2613, 25jaod 369 . . . 4 ((A NC B NC ) → ((¬ A = 0c ¬ B = 0c) → ¬ (A +c B) = 0c))
271, 26syl5bi 208 . . 3 ((A NC B NC ) → (¬ (A = 0c B = 0c) → ¬ (A +c B) = 0c))
2827con4d 97 . 2 ((A NC B NC ) → ((A +c B) = 0c → (A = 0c B = 0c)))
29 addceq12 4385 . . 3 ((A = 0c B = 0c) → (A +c B) = (0c +c 0c))
30 addcid2 4407 . . 3 (0c +c 0c) = 0c
3129, 30syl6eq 2401 . 2 ((A = 0c B = 0c) → (A +c B) = 0c)
3228, 31impbid1 194 1 ((A NC B NC ) → ((A +c B) = 0c ↔ (A = 0c B = 0c)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  1cc1c 4134  0cc0c 4374   +c cplc 4375   NC cncs 6088 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101 This theorem is referenced by:  nnc3n3p1  6278
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