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Mirrors > Home > ILE Home > Th. List > unfiin | Unicode version |
Description: The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.) |
Ref | Expression |
---|---|
unfiin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 518 | . . . . . 6 | |
2 | simpr 109 | . . . . . 6 | |
3 | inss1 3296 | . . . . . . 7 | |
4 | 3 | a1i 9 | . . . . . 6 |
5 | undiffi 6813 | . . . . . 6 | |
6 | 1, 2, 4, 5 | syl3anc 1216 | . . . . 5 |
7 | simplr 519 | . . . . . 6 | |
8 | inss2 3297 | . . . . . . 7 | |
9 | 8 | a1i 9 | . . . . . 6 |
10 | undiffi 6813 | . . . . . 6 | |
11 | 7, 2, 9, 10 | syl3anc 1216 | . . . . 5 |
12 | 6, 11 | uneq12d 3231 | . . . 4 |
13 | unundi 3237 | . . . 4 | |
14 | 12, 13 | syl6eqr 2190 | . . 3 |
15 | diffifi 6788 | . . . . . 6 | |
16 | 1, 2, 4, 15 | syl3anc 1216 | . . . . 5 |
17 | diffifi 6788 | . . . . . 6 | |
18 | 7, 2, 9, 17 | syl3anc 1216 | . . . . 5 |
19 | incom 3268 | . . . . . . . . . 10 | |
20 | 19 | difeq2i 3191 | . . . . . . . . 9 |
21 | difin 3313 | . . . . . . . . 9 | |
22 | 20, 21 | eqtr3i 2162 | . . . . . . . 8 |
23 | 22 | ineq2i 3274 | . . . . . . 7 |
24 | difss 3202 | . . . . . . . 8 | |
25 | disjdif 3435 | . . . . . . . 8 | |
26 | ssdisj 3419 | . . . . . . . 8 | |
27 | 24, 25, 26 | mp2an 422 | . . . . . . 7 |
28 | 23, 27 | eqtri 2160 | . . . . . 6 |
29 | 28 | a1i 9 | . . . . 5 |
30 | unfidisj 6810 | . . . . 5 | |
31 | 16, 18, 29, 30 | syl3anc 1216 | . . . 4 |
32 | difundir 3329 | . . . . . . 7 | |
33 | 32 | ineq2i 3274 | . . . . . 6 |
34 | disjdif 3435 | . . . . . 6 | |
35 | 33, 34 | eqtr3i 2162 | . . . . 5 |
36 | 35 | a1i 9 | . . . 4 |
37 | unfidisj 6810 | . . . 4 | |
38 | 2, 31, 36, 37 | syl3anc 1216 | . . 3 |
39 | 14, 38 | eqeltrd 2216 | . 2 |
40 | 39 | 3impa 1176 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 cdif 3068 cun 3069 cin 3070 wss 3071 c0 3363 cfn 6634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1o 6313 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: (None) |
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