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Mirrors > Home > ILE Home > Th. List > djuassen | Unicode version |
Description: Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
djuassen | ⊔ ⊔ ⊔ ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4055 | . . . . . 6 | |
2 | simp1 981 | . . . . . 6 | |
3 | xpsnen2g 6723 | . . . . . 6 | |
4 | 1, 2, 3 | sylancr 410 | . . . . 5 |
5 | 4 | ensymd 6677 | . . . 4 |
6 | 1oex 6321 | . . . . . . 7 | |
7 | 1 | snex 4109 | . . . . . . . 8 |
8 | simp2 982 | . . . . . . . 8 | |
9 | xpexg 4653 | . . . . . . . 8 | |
10 | 7, 8, 9 | sylancr 410 | . . . . . . 7 |
11 | xpsnen2g 6723 | . . . . . . 7 | |
12 | 6, 10, 11 | sylancr 410 | . . . . . 6 |
13 | xpsnen2g 6723 | . . . . . . 7 | |
14 | 1, 8, 13 | sylancr 410 | . . . . . 6 |
15 | entr 6678 | . . . . . 6 | |
16 | 12, 14, 15 | syl2anc 408 | . . . . 5 |
17 | 16 | ensymd 6677 | . . . 4 |
18 | xp01disjl 6331 | . . . . 5 | |
19 | 18 | a1i 9 | . . . 4 |
20 | djuenun 7068 | . . . 4 ⊔ | |
21 | 5, 17, 19, 20 | syl3anc 1216 | . . 3 ⊔ |
22 | 6 | snex 4109 | . . . . . . 7 |
23 | simp3 983 | . . . . . . 7 | |
24 | xpexg 4653 | . . . . . . 7 | |
25 | 22, 23, 24 | sylancr 410 | . . . . . 6 |
26 | xpsnen2g 6723 | . . . . . 6 | |
27 | 6, 25, 26 | sylancr 410 | . . . . 5 |
28 | xpsnen2g 6723 | . . . . . 6 | |
29 | 6, 23, 28 | sylancr 410 | . . . . 5 |
30 | entr 6678 | . . . . 5 | |
31 | 27, 29, 30 | syl2anc 408 | . . . 4 |
32 | 31 | ensymd 6677 | . . 3 |
33 | indir 3325 | . . . . 5 | |
34 | xp01disjl 6331 | . . . . . . 7 | |
35 | xp01disjl 6331 | . . . . . . . . 9 | |
36 | 35 | xpeq2i 4560 | . . . . . . . 8 |
37 | xpindi 4674 | . . . . . . . 8 | |
38 | xp0 4958 | . . . . . . . 8 | |
39 | 36, 37, 38 | 3eqtr3i 2168 | . . . . . . 7 |
40 | 34, 39 | uneq12i 3228 | . . . . . 6 |
41 | un0 3396 | . . . . . 6 | |
42 | 40, 41 | eqtri 2160 | . . . . 5 |
43 | 33, 42 | eqtri 2160 | . . . 4 |
44 | 43 | a1i 9 | . . 3 |
45 | djuenun 7068 | . . 3 ⊔ ⊔ ⊔ | |
46 | 21, 32, 44, 45 | syl3anc 1216 | . 2 ⊔ ⊔ |
47 | df-dju 6923 | . . . . . 6 ⊔ | |
48 | 47 | xpeq2i 4560 | . . . . 5 ⊔ |
49 | xpundi 4595 | . . . . 5 | |
50 | 48, 49 | eqtri 2160 | . . . 4 ⊔ |
51 | 50 | uneq2i 3227 | . . 3 ⊔ |
52 | df-dju 6923 | . . 3 ⊔ ⊔ ⊔ | |
53 | unass 3233 | . . 3 | |
54 | 51, 52, 53 | 3eqtr4i 2170 | . 2 ⊔ ⊔ |
55 | 46, 54 | breqtrrdi 3970 | 1 ⊔ ⊔ ⊔ ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 962 wceq 1331 wcel 1480 cvv 2686 cun 3069 cin 3070 c0 3363 csn 3527 class class class wbr 3929 cxp 4537 c1o 6306 cen 6632 ⊔ cdju 6922 |
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 |
This theorem depends on definitions: df-bi 116 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-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-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-1st 6038 df-2nd 6039 df-1o 6313 df-er 6429 df-en 6635 df-dju 6923 df-inl 6932 df-inr 6933 |
This theorem is referenced by: (None) |
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