Theorem addceq0 6220

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)))

Proof of Theorem addceq0

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ianor 474	. . . 4 ⊢ (¬ (A = 0c ∧ B = 0c) ↔ (¬ A = 0c ∨ ¬ B = 0c))
2		nc0suc 6218	. . . . . . . 8 ⊢ (A ∈ NC → (A = 0c ∨ ∃p ∈ NC A = (p +c 1c)))
3	2	ord 366	. . . . . . 7 ⊢ (A ∈ NC → (¬ A = 0c → ∃p ∈ NC A = (p +c 1c)))
4	3	adantr 451	. . . . . 6 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (¬ A = 0c → ∃p ∈ NC A = (p +c 1c)))
5		addc32 4417	. . . . . . . . 9 ⊢ ((p +c 1c) +c B) = ((p +c B) +c 1c)
6		0cnsuc 4402	. . . . . . . . 9 ⊢ ((p +c B) +c 1c) ≠ 0c
7	5, 6	eqnetri 2534	. . . . . . . 8 ⊢ ((p +c 1c) +c B) ≠ 0c
8		addceq1 4384	. . . . . . . . . 10 ⊢ (A = (p +c 1c) → (A +c B) = ((p +c 1c) +c B))
9	8	eqeq1d 2361	. . . . . . . . 9 ⊢ (A = (p +c 1c) → ((A +c B) = 0c ↔ ((p +c 1c) +c B) = 0c))
10	9	necon3bbid 2551	. . . . . . . 8 ⊢ (A = (p +c 1c) → (¬ (A +c B) = 0c ↔ ((p +c 1c) +c B) ≠ 0c))
11	7, 10	mpbiri 224	. . . . . . 7 ⊢ (A = (p +c 1c) → ¬ (A +c B) = 0c)
12	11	rexlimivw 2735	. . . . . 6 ⊢ (∃p ∈ NC A = (p +c 1c) → ¬ (A +c B) = 0c)
13	4, 12	syl6 29	. . . . 5 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (¬ A = 0c → ¬ (A +c B) = 0c))
14		nc0suc 6218	. . . . . . . 8 ⊢ (B ∈ NC → (B = 0c ∨ ∃p ∈ NC B = (p +c 1c)))
15	14	ord 366	. . . . . . 7 ⊢ (B ∈ NC → (¬ B = 0c → ∃p ∈ NC B = (p +c 1c)))
16	15	adantl 452	. . . . . 6 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (¬ B = 0c → ∃p ∈ NC B = (p +c 1c)))
17		addcass 4416	. . . . . . . . 9 ⊢ ((A +c p) +c 1c) = (A +c (p +c 1c))
18		0cnsuc 4402	. . . . . . . . 9 ⊢ ((A +c p) +c 1c) ≠ 0c
19	17, 18	eqnetrri 2536	. . . . . . . 8 ⊢ (A +c (p +c 1c)) ≠ 0c
20		addceq2 4385	. . . . . . . . . 10 ⊢ (B = (p +c 1c) → (A +c B) = (A +c (p +c 1c)))
21	20	eqeq1d 2361	. . . . . . . . 9 ⊢ (B = (p +c 1c) → ((A +c B) = 0c ↔ (A +c (p +c 1c)) = 0c))
22	21	necon3bbid 2551	. . . . . . . 8 ⊢ (B = (p +c 1c) → (¬ (A +c B) = 0c ↔ (A +c (p +c 1c)) ≠ 0c))
23	19, 22	mpbiri 224	. . . . . . 7 ⊢ (B = (p +c 1c) → ¬ (A +c B) = 0c)
24	23	rexlimivw 2735	. . . . . 6 ⊢ (∃p ∈ NC B = (p +c 1c) → ¬ (A +c B) = 0c)
25	16, 24	syl6 29	. . . . 5 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (¬ B = 0c → ¬ (A +c B) = 0c))
26	13, 25	jaod 369	. . . 4 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((¬ A = 0c ∨ ¬ B = 0c) → ¬ (A +c B) = 0c))
27	1, 26	syl5bi 208	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (¬ (A = 0c ∧ B = 0c) → ¬ (A +c B) = 0c))
28	27	con4d 97	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A +c B) = 0c → (A = 0c ∧ B = 0c)))
29		addceq12 4386	. . 3 ⊢ ((A = 0c ∧ B = 0c) → (A +c B) = (0c +c 0c))
30		addcid2 4408	. . 3 ⊢ (0c +c 0c) = 0c
31	29, 30	syl6eq 2401	. 2 ⊢ ((A = 0c ∧ B = 0c) → (A +c B) = 0c)
32	28, 31	impbid1 194	1 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A +c B) = 0c ↔ (A = 0c ∧ B = 0c)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  1_cc1c 4135  0_cc0c 4375   +_c cplc 4376   NC cncs 6089

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-nc 6102

This theorem is referenced by:  nnc3n3p1  6279