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Mirrors > Home > ILE Home > Th. List > Mathboxes > nninfsellemsuc | Unicode version |
Description: Lemma for nninfself 13627. (Contributed by Jim Kingdon, 6-Aug-2022.) |
Ref | Expression |
---|---|
nninfsellemsuc | ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2 4555 | . . . . 5 | |
2 | nninfsellemcl 13625 | . . . . . 6 ℕ∞ | |
3 | el2oss1o 6391 | . . . . . 6 | |
4 | 2, 3 | syl 14 | . . . . 5 ℕ∞ |
5 | 1, 4 | sylan2 284 | . . . 4 ℕ∞ |
6 | 5 | adantr 274 | . . 3 ℕ∞ |
7 | iftrue 3510 | . . . 4 | |
8 | 7 | adantl 275 | . . 3 ℕ∞ |
9 | 6, 8 | sseqtrrd 3167 | . 2 ℕ∞ |
10 | simpl 108 | . . . . . . 7 | |
11 | 10 | con3i 622 | . . . . . 6 |
12 | df-suc 4332 | . . . . . . . 8 | |
13 | 12 | raleqi 2656 | . . . . . . 7 |
14 | ralunb 3288 | . . . . . . 7 | |
15 | 13, 14 | bitri 183 | . . . . . 6 |
16 | 11, 15 | sylnibr 667 | . . . . 5 |
17 | 16 | iffalsed 3515 | . . . 4 |
18 | 0ss 3432 | . . . 4 | |
19 | 17, 18 | eqsstrdi 3180 | . . 3 |
20 | 19 | adantl 275 | . 2 ℕ∞ |
21 | nninfsellemdc 13624 | . . 3 ℕ∞ DECID | |
22 | exmiddc 822 | . . 3 DECID | |
23 | 21, 22 | syl 14 | . 2 ℕ∞ |
24 | 9, 20, 23 | mpjaodan 788 | 1 ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 820 wceq 1335 wcel 2128 wral 2435 cun 3100 wss 3102 c0 3394 cif 3505 csn 3560 cmpt 4026 csuc 4326 com 4550 cfv 5171 (class class class)co 5825 c1o 6357 c2o 6358 cmap 6594 ℕ∞xnninf 7064 |
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-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
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-rab 2444 df-v 2714 df-sbc 2938 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 3774 df-int 3809 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1o 6364 df-2o 6365 df-map 6596 df-nninf 7065 |
This theorem is referenced by: nninfself 13627 |
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