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Mirrors > Home > NFE Home > Th. List > addceq0 | Unicode version |
Description: The sum of two cardinals is zero iff both addends are zero. (Contributed by SF, 12-Mar-2015.) |
Ref | Expression |
---|---|
addceq0 | NC NC 0c 0c 0c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ianor 474 | . . . 4 0c 0c 0c 0c | |
2 | nc0suc 6217 | . . . . . . . 8 NC 0c NC 1c | |
3 | 2 | ord 366 | . . . . . . 7 NC 0c NC 1c |
4 | 3 | adantr 451 | . . . . . 6 NC NC 0c NC 1c |
5 | addc32 4416 | . . . . . . . . 9 1c 1c | |
6 | 0cnsuc 4401 | . . . . . . . . 9 1c 0c | |
7 | 5, 6 | eqnetri 2533 | . . . . . . . 8 1c 0c |
8 | addceq1 4383 | . . . . . . . . . 10 1c 1c | |
9 | 8 | eqeq1d 2361 | . . . . . . . . 9 1c 0c 1c 0c |
10 | 9 | necon3bbid 2550 | . . . . . . . 8 1c 0c 1c 0c |
11 | 7, 10 | mpbiri 224 | . . . . . . 7 1c 0c |
12 | 11 | rexlimivw 2734 | . . . . . 6 NC 1c 0c |
13 | 4, 12 | syl6 29 | . . . . 5 NC NC 0c 0c |
14 | nc0suc 6217 | . . . . . . . 8 NC 0c NC 1c | |
15 | 14 | ord 366 | . . . . . . 7 NC 0c NC 1c |
16 | 15 | adantl 452 | . . . . . 6 NC NC 0c NC 1c |
17 | addcass 4415 | . . . . . . . . 9 1c 1c | |
18 | 0cnsuc 4401 | . . . . . . . . 9 1c 0c | |
19 | 17, 18 | eqnetrri 2535 | . . . . . . . 8 1c 0c |
20 | addceq2 4384 | . . . . . . . . . 10 1c 1c | |
21 | 20 | eqeq1d 2361 | . . . . . . . . 9 1c 0c 1c 0c |
22 | 21 | necon3bbid 2550 | . . . . . . . 8 1c 0c 1c 0c |
23 | 19, 22 | mpbiri 224 | . . . . . . 7 1c 0c |
24 | 23 | rexlimivw 2734 | . . . . . 6 NC 1c 0c |
25 | 16, 24 | syl6 29 | . . . . 5 NC NC 0c 0c |
26 | 13, 25 | jaod 369 | . . . 4 NC NC 0c 0c 0c |
27 | 1, 26 | syl5bi 208 | . . 3 NC NC 0c 0c 0c |
28 | 27 | con4d 97 | . 2 NC NC 0c 0c 0c |
29 | addceq12 4385 | . . 3 0c 0c 0c 0c | |
30 | addcid2 4407 | . . 3 0c 0c 0c | |
31 | 29, 30 | syl6eq 2401 | . 2 0c 0c 0c |
32 | 28, 31 | impbid1 194 | 1 NC NC 0c 0c 0c |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wo 357 wa 358 wceq 1642 wcel 1710 wne 2516 wrex 2615 1cc1c 4134 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-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-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-txp 5736 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-lec 6099 df-nc 6101 |
This theorem is referenced by: nnc3n3p1 6278 |
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