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Mirrors > Home > MPE Home > Th. List > iunctb | Structured version Visualization version Unicode version |
Description: The countable union of countable sets is countable (indexed union version of unictb 9397). (Contributed by Mario Carneiro, 18-Jan-2014.) |
Ref | Expression |
---|---|
iunctb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 | |
2 | simpl 473 | . . . 4 | |
3 | reldom 7961 | . . . . . . . 8 | |
4 | 3 | brrelexi 5158 | . . . . . . 7 |
5 | 4 | adantr 481 | . . . . . 6 |
6 | ovex 6678 | . . . . . . 7 | |
7 | 6 | rgenw 2924 | . . . . . 6 |
8 | iunexg 7143 | . . . . . 6 | |
9 | 5, 7, 8 | sylancl 694 | . . . . 5 |
10 | acncc 9262 | . . . . 5 AC | |
11 | 9, 10 | syl6eleqr 2712 | . . . 4 AC |
12 | acndom 8874 | . . . 4 AC AC | |
13 | 2, 11, 12 | sylc 65 | . . 3 AC |
14 | simpr 477 | . . 3 | |
15 | omex 8540 | . . . . . 6 | |
16 | xpdom1g 8057 | . . . . . 6 | |
17 | 15, 2, 16 | sylancr 695 | . . . . 5 |
18 | xpomen 8838 | . . . . 5 | |
19 | domentr 8015 | . . . . 5 | |
20 | 17, 18, 19 | sylancl 694 | . . . 4 |
21 | 3 | brrelexi 5158 | . . . . . . 7 |
22 | 21 | ralimi 2952 | . . . . . 6 |
23 | iunexg 7143 | . . . . . 6 | |
24 | 4, 22, 23 | syl2an 494 | . . . . 5 |
25 | omelon 8543 | . . . . . 6 | |
26 | onenon 8775 | . . . . . 6 | |
27 | 25, 26 | ax-mp 5 | . . . . 5 |
28 | numacn 8872 | . . . . 5 AC | |
29 | 24, 27, 28 | mpisyl 21 | . . . 4 AC |
30 | acndom2 8877 | . . . 4 AC AC | |
31 | 20, 29, 30 | sylc 65 | . . 3 AC |
32 | 1, 13, 14, 31 | iundomg 9363 | . 2 |
33 | domtr 8009 | . 2 | |
34 | 32, 20, 33 | syl2anc 693 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wcel 1990 wral 2912 cvv 3200 csn 4177 ciun 4520 class class class wbr 4653 cxp 5112 cdm 5114 con0 5723 (class class class)co 6650 com 7065 cmap 7857 cen 7952 cdom 7953 ccrd 8761 AC wacn 8764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-card 8765 df-acn 8768 |
This theorem is referenced by: unictb 9397 heiborlem3 33612 |
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