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Mirrors > Home > ILE Home > Th. List > iunfidisj | Unicode version |
Description: The finite union of disjoint finite sets is finite. Note that depends on , i.e. can be thought of as . (Contributed by NM, 23-Mar-2006.) (Revised by Jim Kingdon, 7-Oct-2022.) |
Ref | Expression |
---|---|
iunfidisj | Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq1 3862 | . . 3 | |
2 | 1 | eleq1d 2226 | . 2 |
3 | iuneq1 3862 | . . 3 | |
4 | 3 | eleq1d 2226 | . 2 |
5 | iuneq1 3862 | . . 3 | |
6 | 5 | eleq1d 2226 | . 2 |
7 | iuneq1 3862 | . . 3 | |
8 | 7 | eleq1d 2226 | . 2 |
9 | 0iun 3906 | . . . 4 | |
10 | 0fin 6829 | . . . 4 | |
11 | 9, 10 | eqeltri 2230 | . . 3 |
12 | 11 | a1i 9 | . 2 Disj |
13 | iunxun 3928 | . . . 4 | |
14 | simpr 109 | . . . . 5 Disj | |
15 | nfcsb1v 3064 | . . . . . . . 8 | |
16 | csbeq1a 3040 | . . . . . . . 8 | |
17 | 15, 16 | iunxsngf 3926 | . . . . . . 7 |
18 | 17 | elv 2716 | . . . . . 6 |
19 | simplrr 526 | . . . . . . . 8 Disj | |
20 | 19 | eldifad 3113 | . . . . . . 7 Disj |
21 | simpll2 1022 | . . . . . . 7 Disj | |
22 | 15 | nfel1 2310 | . . . . . . . 8 |
23 | 16 | eleq1d 2226 | . . . . . . . 8 |
24 | 22, 23 | rspc 2810 | . . . . . . 7 |
25 | 20, 21, 24 | sylc 62 | . . . . . 6 Disj |
26 | 18, 25 | eqeltrid 2244 | . . . . 5 Disj |
27 | simpll3 1023 | . . . . . 6 Disj Disj | |
28 | simplrl 525 | . . . . . 6 Disj | |
29 | 20 | snssd 3701 | . . . . . 6 Disj |
30 | 19 | eldifbd 3114 | . . . . . . 7 Disj |
31 | disjsn 3621 | . . . . . . 7 | |
32 | 30, 31 | sylibr 133 | . . . . . 6 Disj |
33 | disjiun 3960 | . . . . . 6 Disj | |
34 | 27, 28, 29, 32, 33 | syl13anc 1222 | . . . . 5 Disj |
35 | unfidisj 6866 | . . . . 5 | |
36 | 14, 26, 34, 35 | syl3anc 1220 | . . . 4 Disj |
37 | 13, 36 | eqeltrid 2244 | . . 3 Disj |
38 | 37 | ex 114 | . 2 Disj |
39 | simp1 982 | . 2 Disj | |
40 | 2, 4, 6, 8, 12, 38, 39 | findcard2d 6836 | 1 Disj |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 wral 2435 cvv 2712 csb 3031 cdif 3099 cun 3100 cin 3101 wss 3102 c0 3394 csn 3560 ciun 3849 Disj wdisj 3942 cfn 6685 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-disj 3943 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1o 6363 df-er 6480 df-en 6686 df-fin 6688 |
This theorem is referenced by: fsum2dlemstep 11331 fisumcom2 11335 fsumiun 11374 hashiun 11375 hash2iun 11376 fprod2dlemstep 11519 fprodcom2fi 11523 |
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