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Mirrors > Home > ILE Home > Th. List > Mathboxes > nninfsellemeq | Unicode version |
Description: Lemma for nninfsel 13738. (Contributed by Jim Kingdon, 9-Aug-2022.) |
Ref | Expression |
---|---|
nninfsel.e | ℕ∞ |
nninfsel.q | ℕ∞ |
nninfsel.1 | |
nninfsel.n | |
nninfsel.qk | |
nninfsel.qn |
Ref | Expression |
---|---|
nninfsellemeq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfsel.e | . . . . 5 ℕ∞ | |
2 | 1 | nninfself 13734 | . . . 4 ℕ∞ℕ∞ |
3 | 2 | a1i 9 | . . 3 ℕ∞ℕ∞ |
4 | nninfsel.q | . . 3 ℕ∞ | |
5 | 3, 4 | ffvelrnd 5615 | . 2 ℕ∞ |
6 | nninfsel.n | . 2 | |
7 | fveq1 5479 | . . . . . . . . . . 11 | |
8 | 7 | eqeq1d 2173 | . . . . . . . . . 10 |
9 | 8 | ralbidv 2464 | . . . . . . . . 9 |
10 | 9 | ifbid 3536 | . . . . . . . 8 |
11 | 10 | mpteq2dv 4067 | . . . . . . 7 |
12 | omex 4564 | . . . . . . . 8 | |
13 | 12 | mptex 5705 | . . . . . . 7 |
14 | 11, 1, 13 | fvmpt 5557 | . . . . . 6 ℕ∞ |
15 | 4, 14 | syl 14 | . . . . 5 |
16 | 15 | adantr 274 | . . . 4 |
17 | simpr 109 | . . . . . . . 8 | |
18 | simplr 520 | . . . . . . . 8 | |
19 | 17, 18 | eqeltrd 2241 | . . . . . . 7 |
20 | nnord 4583 | . . . . . . . . 9 | |
21 | vex 2724 | . . . . . . . . . 10 | |
22 | ordelsuc 4476 | . . . . . . . . . 10 | |
23 | 21, 22 | mpan 421 | . . . . . . . . 9 |
24 | 6, 20, 23 | 3syl 17 | . . . . . . . 8 |
25 | 24 | ad2antrr 480 | . . . . . . 7 |
26 | 19, 25 | mpbid 146 | . . . . . 6 |
27 | nninfsel.qk | . . . . . . 7 | |
28 | 27 | ad2antrr 480 | . . . . . 6 |
29 | ssralv 3201 | . . . . . 6 | |
30 | 26, 28, 29 | sylc 62 | . . . . 5 |
31 | 30 | iftrued 3522 | . . . 4 |
32 | simpr 109 | . . . . 5 | |
33 | 6 | adantr 274 | . . . . 5 |
34 | elnn 4577 | . . . . 5 | |
35 | 32, 33, 34 | syl2anc 409 | . . . 4 |
36 | 1onn 6479 | . . . . 5 | |
37 | 36 | a1i 9 | . . . 4 |
38 | 16, 31, 35, 37 | fvmptd 5561 | . . 3 |
39 | 38 | ralrimiva 2537 | . 2 |
40 | 21 | sucid 4389 | . . . . . . 7 |
41 | 40 | a1i 9 | . . . . . 6 |
42 | 1n0 6391 | . . . . . . . 8 | |
43 | 42 | nesymi 2380 | . . . . . . 7 |
44 | simpr 109 | . . . . . . . . . . . . 13 | |
45 | 44 | eleq2d 2234 | . . . . . . . . . . . 12 |
46 | 45 | ifbid 3536 | . . . . . . . . . . 11 |
47 | 46 | mpteq2dv 4067 | . . . . . . . . . 10 |
48 | 47 | fveq2d 5484 | . . . . . . . . 9 |
49 | nninfsel.qn | . . . . . . . . . 10 | |
50 | 49 | adantr 274 | . . . . . . . . 9 |
51 | 48, 50 | eqtrd 2197 | . . . . . . . 8 |
52 | 51 | eqeq1d 2173 | . . . . . . 7 |
53 | 43, 52 | mtbiri 665 | . . . . . 6 |
54 | elequ2 2140 | . . . . . . . . . . . 12 | |
55 | 54 | ifbid 3536 | . . . . . . . . . . 11 |
56 | 55 | mpteq2dv 4067 | . . . . . . . . . 10 |
57 | 56 | fveq2d 5484 | . . . . . . . . 9 |
58 | 57 | eqeq1d 2173 | . . . . . . . 8 |
59 | 58 | notbid 657 | . . . . . . 7 |
60 | 59 | rspcev 2825 | . . . . . 6 |
61 | 41, 53, 60 | syl2anc 409 | . . . . 5 |
62 | rexnalim 2453 | . . . . 5 | |
63 | 61, 62 | syl 14 | . . . 4 |
64 | 63 | iffalsed 3525 | . . 3 |
65 | peano1 4565 | . . . 4 | |
66 | 65 | a1i 9 | . . 3 |
67 | 15, 64, 6, 66 | fvmptd 5561 | . 2 |
68 | 5, 6, 39, 67 | nnnninfeq 7083 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 wrex 2443 cvv 2721 wss 3111 c0 3404 cif 3515 cmpt 4037 word 4334 csuc 4337 com 4561 wf 5178 cfv 5182 (class class class)co 5836 c1o 6368 c2o 6369 cmap 6605 ℕ∞xnninf 7075 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1o 6375 df-2o 6376 df-map 6607 df-nninf 7076 |
This theorem is referenced by: nninfsellemqall 13736 |
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