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Mirrors > Home > NFE Home > Th. List > addceq2 | Unicode version |
Description: Equality law for cardinal addition. (Contributed by SF, 15-Jan-2015.) |
Ref | Expression |
---|---|
addceq2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw1eq 4143 | . . . . 5 1 1 | |
2 | pw1eq 4143 | . . . . 5 1 1 1 1 1 1 | |
3 | 1, 2 | syl 15 | . . . 4 1 1 1 1 |
4 | 3 | imakeq2d 4229 | . . 3 Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 |
5 | 4 | imakeq1d 4228 | . 2 Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 k Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 k |
6 | dfaddc2 4381 | . 2 Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 k | |
7 | dfaddc2 4381 | . 2 Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 k | |
8 | 5, 6, 7 | 3eqtr4g 2410 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1642 ∼ ccompl 3205 cdif 3206 cun 3207 cin 3208 csymdif 3209 1cc1c 4134 1 cpw1 4135 Ins2k cins2k 4176 Ins3k cins3k 4177 kcimak 4179 SIk csik 4181 Sk cssetk 4183 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 addceq2i 4387 addceq2d 4390 nncaddccl 4419 lefinaddc 4450 addcnnul 4453 preaddccan2lem1 4454 preaddccan2 4455 nulge 4456 leltfintr 4458 ltfintr 4459 ltfinp1 4462 lefinlteq 4463 lefinrflx 4467 ltlefin 4468 tfinltfinlem1 4500 eventfin 4517 oddtfin 4518 sfinltfin 4535 braddcfn 5826 dflec2 6210 addceq0 6219 tlecg 6230 nclenn 6249 csucex 6259 addccan2nclem2 6264 addccan2nc 6265 ncslesuc 6267 nchoicelem17 6305 |
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