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Mirrors > Home > ILE Home > Th. List > Mathboxes > nninfsel | Unicode version |
Description: is a selection function for ℕ∞. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.) |
Ref | Expression |
---|---|
nninfsel.e | ℕ∞ |
nninfsel.q | ℕ∞ |
nninfsel.1 |
Ref | Expression |
---|---|
nninfsel | ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfsel.q | . 2 ℕ∞ | |
2 | nninfsel.e | . . . . 5 ℕ∞ | |
3 | nninfsel.1 | . . . . 5 | |
4 | 2, 1, 3 | nninfsellemeqinf 13215 | . . . 4 |
5 | 4 | fveq2d 5425 | . . 3 |
6 | 5, 3 | eqtr3d 2174 | . 2 |
7 | 1 | adantr 274 | . . . 4 ℕ∞ |
8 | 3 | adantr 274 | . . . 4 |
9 | simpr 109 | . . . 4 | |
10 | 2, 7, 8, 9 | nninfsellemqall 13214 | . . 3 |
11 | 10 | ralrimiva 2505 | . 2 |
12 | 1, 6, 11 | nninfall 13207 | 1 ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 c0 3363 cif 3474 cmpt 3989 csuc 4287 com 4504 cfv 5123 (class class class)co 5774 c1o 6306 c2o 6307 cmap 6542 ℕ∞xnninf 7005 |
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 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 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-if 3475 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-iom 4505 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-ov 5777 df-oprab 5778 df-mpo 5779 df-1o 6313 df-2o 6314 df-map 6544 df-nninf 7007 |
This theorem is referenced by: nninfomnilem 13217 |
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