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Mirrors > Home > NFE Home > Th. List > addccan2nc | Unicode version |
Description: Cancellation law for addition over the cardinal numbers. Biconditional form of theorem XI.3.2 of [Rosser] p. 391. (Contributed by Scott Fenton, 2-Aug-2019.) |
Ref | Expression |
---|---|
addccan2nc | Nn NC NC |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addccan2nclem2 6264 | . . . 4 NC NC | |
2 | addceq1 4383 | . . . . . 6 0c 0c | |
3 | addceq1 4383 | . . . . . 6 0c 0c | |
4 | 2, 3 | eqeq12d 2367 | . . . . 5 0c 0c 0c |
5 | 4 | imbi1d 308 | . . . 4 0c 0c 0c |
6 | addceq1 4383 | . . . . . 6 | |
7 | addceq1 4383 | . . . . . 6 | |
8 | 6, 7 | eqeq12d 2367 | . . . . 5 |
9 | 8 | imbi1d 308 | . . . 4 |
10 | addceq1 4383 | . . . . . 6 1c 1c | |
11 | addceq1 4383 | . . . . . 6 1c 1c | |
12 | 10, 11 | eqeq12d 2367 | . . . . 5 1c 1c 1c |
13 | 12 | imbi1d 308 | . . . 4 1c 1c 1c |
14 | addceq1 4383 | . . . . . 6 | |
15 | addceq1 4383 | . . . . . 6 | |
16 | 14, 15 | eqeq12d 2367 | . . . . 5 |
17 | 16 | imbi1d 308 | . . . 4 |
18 | addcid2 4407 | . . . . . . 7 0c | |
19 | addcid2 4407 | . . . . . . 7 0c | |
20 | 18, 19 | eqeq12i 2366 | . . . . . 6 0c 0c |
21 | 20 | biimpi 186 | . . . . 5 0c 0c |
22 | 21 | a1i 10 | . . . 4 NC NC 0c 0c |
23 | addc32 4416 | . . . . . . 7 1c 1c | |
24 | addc32 4416 | . . . . . . 7 1c 1c | |
25 | 23, 24 | eqeq12i 2366 | . . . . . 6 1c 1c 1c 1c |
26 | nnnc 6146 | . . . . . . . . . . 11 Nn NC | |
27 | ncaddccl 6144 | . . . . . . . . . . 11 NC NC NC | |
28 | 26, 27 | sylan 457 | . . . . . . . . . 10 Nn NC NC |
29 | 28 | adantrr 697 | . . . . . . . . 9 Nn NC NC NC |
30 | 29 | adantr 451 | . . . . . . . 8 Nn NC NC NC |
31 | ncaddccl 6144 | . . . . . . . . . . 11 NC NC NC | |
32 | 26, 31 | sylan 457 | . . . . . . . . . 10 Nn NC NC |
33 | 32 | adantrl 696 | . . . . . . . . 9 Nn NC NC NC |
34 | 33 | adantr 451 | . . . . . . . 8 Nn NC NC NC |
35 | peano4nc 6150 | . . . . . . . . 9 NC NC 1c 1c | |
36 | 35 | biimpd 198 | . . . . . . . 8 NC NC 1c 1c |
37 | 30, 34, 36 | syl2anc 642 | . . . . . . 7 Nn NC NC 1c 1c |
38 | simpr 447 | . . . . . . 7 Nn NC NC | |
39 | 37, 38 | syld 40 | . . . . . 6 Nn NC NC 1c 1c |
40 | 25, 39 | syl5bi 208 | . . . . 5 Nn NC NC 1c 1c |
41 | 40 | ex 423 | . . . 4 Nn NC NC 1c 1c |
42 | 1, 5, 9, 13, 17, 22, 41 | findsd 4410 | . . 3 Nn NC NC |
43 | 42 | 3impb 1147 | . 2 Nn NC NC |
44 | addceq2 4384 | . 2 | |
45 | 43, 44 | impbid1 194 | 1 Nn NC NC |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 w3a 934 wceq 1642 wcel 1710 cvv 2859 1cc1c 4134 Nn cnnc 4373 0cc0c 4374 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-csb 3137 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-iun 3971 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-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-txp 5736 df-fix 5740 df-cup 5742 df-disj 5744 df-addcfn 5746 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-nc 6101 |
This theorem is referenced by: lecadd2 6266 |
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