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Mirrors > Home > NFE Home > Th. List > addcnnul | Unicode version |
Description: If cardinal addition is non-empty, then both addends are non-empty. Theorem X.1.20 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.) |
Ref | Expression |
---|---|
addcnnul |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addceq1 4383 | . . . 4 | |
2 | addccom 4406 | . . . . 5 | |
3 | addcnul1 4452 | . . . . 5 | |
4 | 2, 3 | eqtri 2373 | . . . 4 |
5 | 1, 4 | syl6eq 2401 | . . 3 |
6 | 5 | necon3i 2555 | . 2 |
7 | addceq2 4384 | . . . 4 | |
8 | addcnul1 4452 | . . . 4 | |
9 | 7, 8 | syl6eq 2401 | . . 3 |
10 | 9 | necon3i 2555 | . 2 |
11 | 6, 10 | jca 518 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 wceq 1642 wne 2516 c0 3550 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-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-ins2k 4187 df-ins3k 4188 df-imak 4189 df-sik 4192 df-ssetk 4193 df-addc 4378 |
This theorem is referenced by: preaddccan2 4455 leltfintr 4458 ltfintri 4466 tfinltfinlem1 4500 evenoddnnnul 4514 evenodddisj 4516 oddtfin 4518 |
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