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Mirrors > Home > ILE Home > Th. List > 0ct | Unicode version |
Description: The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.) |
Ref | Expression |
---|---|
0ct | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1o 6337 | . . . . 5 | |
2 | djurcl 6937 | . . . . 5 inr ⊔ | |
3 | 1, 2 | ax-mp 5 | . . . 4 inr ⊔ |
4 | 3 | fconst6 5322 | . . 3 inr ⊔ |
5 | peano1 4508 | . . . . 5 | |
6 | rex0 3380 | . . . . . . . . 9 inl | |
7 | djur 6954 | . . . . . . . . . . 11 ⊔ inl inr | |
8 | 7 | biimpi 119 | . . . . . . . . . 10 ⊔ inl inr |
9 | 8 | ord 713 | . . . . . . . . 9 ⊔ inl inr |
10 | 6, 9 | mpi 15 | . . . . . . . 8 ⊔ inr |
11 | df1o2 6326 | . . . . . . . . 9 | |
12 | 11 | rexeqi 2631 | . . . . . . . 8 inr inr |
13 | 10, 12 | sylib 121 | . . . . . . 7 ⊔ inr |
14 | 0ex 4055 | . . . . . . . 8 | |
15 | fveq2 5421 | . . . . . . . . 9 inr inr | |
16 | 15 | eqeq2d 2151 | . . . . . . . 8 inr inr |
17 | 14, 16 | rexsn 3568 | . . . . . . 7 inr inr |
18 | 13, 17 | sylib 121 | . . . . . 6 ⊔ inr |
19 | 3 | elexi 2698 | . . . . . . . 8 inr |
20 | 19 | fvconst2 5636 | . . . . . . 7 inr inr |
21 | 5, 20 | ax-mp 5 | . . . . . 6 inr inr |
22 | 18, 21 | syl6eqr 2190 | . . . . 5 ⊔ inr |
23 | fveq2 5421 | . . . . . 6 inr inr | |
24 | 23 | rspceeqv 2807 | . . . . 5 inr inr |
25 | 5, 22, 24 | sylancr 410 | . . . 4 ⊔ inr |
26 | 25 | rgen 2485 | . . 3 ⊔ inr |
27 | dffo3 5567 | . . 3 inr ⊔ inr ⊔ ⊔ inr | |
28 | 4, 26, 27 | mpbir2an 926 | . 2 inr ⊔ |
29 | omex 4507 | . . . 4 | |
30 | 19 | snex 4109 | . . . 4 inr |
31 | 29, 30 | xpex 4654 | . . 3 inr |
32 | foeq1 5341 | . . 3 inr ⊔ inr ⊔ | |
33 | 31, 32 | spcev 2780 | . 2 inr ⊔ ⊔ |
34 | 28, 33 | ax-mp 5 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wn 3 wo 697 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 c0 3363 csn 3527 com 4504 cxp 4537 wf 5119 wfo 5121 cfv 5123 c1o 6306 ⊔ cdju 6922 inlcinl 6930 inrcinr 6931 |
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-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
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-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 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-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-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-dju 6923 df-inl 6932 df-inr 6933 |
This theorem is referenced by: enumct 7000 |
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