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Mirrors > Home > ILE Home > Th. List > unfidisj | Unicode version |
Description: The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.) |
Ref | Expression |
---|---|
unfidisj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq2 3265 | . . 3 | |
2 | 1 | eleq1d 2233 | . 2 |
3 | uneq2 3265 | . . 3 | |
4 | 3 | eleq1d 2233 | . 2 |
5 | uneq2 3265 | . . 3 | |
6 | 5 | eleq1d 2233 | . 2 |
7 | uneq2 3265 | . . 3 | |
8 | 7 | eleq1d 2233 | . 2 |
9 | un0 3437 | . . 3 | |
10 | simp1 986 | . . 3 | |
11 | 9, 10 | eqeltrid 2251 | . 2 |
12 | unass 3274 | . . . 4 | |
13 | simpr 109 | . . . . 5 | |
14 | vex 2724 | . . . . . 6 | |
15 | 14 | a1i 9 | . . . . 5 |
16 | simplrr 526 | . . . . . . . . 9 | |
17 | 16 | eldifad 3122 | . . . . . . . 8 |
18 | simp3 988 | . . . . . . . . 9 | |
19 | 18 | ad3antrrr 484 | . . . . . . . 8 |
20 | minel 3465 | . . . . . . . 8 | |
21 | 17, 19, 20 | syl2anc 409 | . . . . . . 7 |
22 | 16 | eldifbd 3123 | . . . . . . 7 |
23 | ioran 742 | . . . . . . 7 | |
24 | 21, 22, 23 | sylanbrc 414 | . . . . . 6 |
25 | elun 3258 | . . . . . 6 | |
26 | 24, 25 | sylnibr 667 | . . . . 5 |
27 | unsnfi 6875 | . . . . 5 | |
28 | 13, 15, 26, 27 | syl3anc 1227 | . . . 4 |
29 | 12, 28 | eqeltrrid 2252 | . . 3 |
30 | 29 | ex 114 | . 2 |
31 | simp2 987 | . 2 | |
32 | 2, 4, 6, 8, 11, 30, 31 | findcard2sd 6849 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 w3a 967 wceq 1342 wcel 2135 cvv 2721 cdif 3108 cun 3109 cin 3110 wss 3111 c0 3404 csn 3570 cfn 6697 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1o 6375 df-er 6492 df-en 6698 df-fin 6700 |
This theorem is referenced by: unfiin 6882 prfidisj 6883 tpfidisj 6884 xpfi 6886 iunfidisj 6902 hashunlem 10706 hashun 10707 fsumsplitsnun 11346 fsum2dlemstep 11361 fsumconst 11381 fprodsplitsn 11560 |
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