New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > addceq1 | GIF version |
Description: Equality law for cardinal addition. (Contributed by SF, 15-Jan-2015.) |
Ref | Expression |
---|---|
addceq1 | ⊢ (A = B → (A +c C) = (B +c C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imakeq2 4225 | . 2 ⊢ (A = B → ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1C) “k A) = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1C) “k B)) | |
2 | dfaddc2 4381 | . 2 ⊢ (A +c C) = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1C) “k A) | |
3 | dfaddc2 4381 | . 2 ⊢ (B +c C) = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1C) “k B) | |
4 | 1, 2, 3 | 3eqtr4g 2410 | 1 ⊢ (A = B → (A +c C) = (B +c C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∼ ccompl 3205 ∖ cdif 3206 ∪ cun 3207 ∩ cin 3208 ⊕ csymdif 3209 1cc1c 4134 ℘1cpw1 4135 Ins2k cins2k 4176 Ins3k cins3k 4177 “k cimak 4179 SIk csik 4181 Sk cssetk 4183 +c cplc 4375 |
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-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-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: addceq12 4385 addceq1i 4386 addceq1d 4389 peano2 4403 nnc0suc 4412 nncaddccl 4419 nnsucelr 4428 nndisjeq 4429 opklefing 4448 opkltfing 4449 addcnnul 4453 preaddccan2 4455 leltfintr 4458 ltfintr 4459 ltfinex 4464 sucevenodd 4510 sucoddeven 4511 evenodddisjlem1 4515 oddtfin 4518 suc11nnc 4558 phi11lem1 4595 phialllem1 4616 braddcfn 5826 dfnnc3 5885 peano4nc 6150 dflec2 6210 lectr 6211 leaddc1 6214 addceq0 6219 csucex 6259 brcsuc 6260 nnltp1c 6262 addccan2nc 6265 nncdiv3lem1 6275 nncdiv3lem2 6276 nncdiv3 6277 nnc3n3p1 6278 nchoicelem9 6297 nchoicelem16 6304 nchoicelem17 6305 fnfreclem2 6318 fnfreclem3 6319 frecsuc 6322 |
Copyright terms: Public domain | W3C validator |