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Mirrors > Home > NFE Home > Th. List > addcexg | Unicode version |
Description: The cardinal sum of two sets is a set. (Contributed by SF, 15-Jan-2015.) |
Ref | Expression |
---|---|
addcexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 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 | |
2 | pw1exg 4302 | . . . . 5 1 | |
3 | pw1exg 4302 | . . . . 5 1 1 1 | |
4 | addcexlem 4382 | . . . . . 6 Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1c | |
5 | imakexg 4299 | . . . . . 6 Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1c 1 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 | |
6 | 4, 5 | mpan 651 | . . . . 5 1 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 |
7 | 2, 3, 6 | 3syl 18 | . . . 4 Ins3k ∼ Ins3k Sk Ins2k Sk k1 1 1c Ins2k Ins2k Sk Ins2k Ins3k Sk Ins3k SIk SIk Sk k1 1 1 1 1ck1 1 |
8 | imakexg 4299 | . . . 4 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 k | |
9 | 7, 8 | sylan 457 | . . 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 k |
10 | 9 | ancoms 439 | . 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 |
11 | 1, 10 | syl5eqel 2437 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 wcel 1710 cvv 2859 ∼ 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-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-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-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-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-p6 4191 df-sik 4192 df-ssetk 4193 df-addc 4378 |
This theorem is referenced by: addcex 4394 lefinaddc 4450 leltfintr 4458 ltfinp1 4462 sucevenodd 4510 sucoddeven 4511 addlec 6208 fnfreclem3 6319 |
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