Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > 0nninf | GIF version |
Description: The zero element of ℕ∞ (the constant sequence equal to ∅). (Contributed by Jim Kingdon, 14-Jul-2022.) |
Ref | Expression |
---|---|
0nninf | ⊢ (ω × {∅}) ∈ ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt2o 6338 | . . . 4 ⊢ ∅ ∈ 2o | |
2 | 1 | fconst6 5322 | . . 3 ⊢ (ω × {∅}):ω⟶2o |
3 | 2onn 6417 | . . . . 5 ⊢ 2o ∈ ω | |
4 | 3 | elexi 2698 | . . . 4 ⊢ 2o ∈ V |
5 | omex 4507 | . . . 4 ⊢ ω ∈ V | |
6 | 4, 5 | elmap 6571 | . . 3 ⊢ ((ω × {∅}) ∈ (2o ↑𝑚 ω) ↔ (ω × {∅}):ω⟶2o) |
7 | 2, 6 | mpbir 145 | . 2 ⊢ (ω × {∅}) ∈ (2o ↑𝑚 ω) |
8 | peano2 4509 | . . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω) | |
9 | 0ex 4055 | . . . . . . 7 ⊢ ∅ ∈ V | |
10 | 9 | fvconst2 5636 | . . . . . 6 ⊢ (suc 𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ∅) |
11 | 8, 10 | syl 14 | . . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ∅) |
12 | 9 | fvconst2 5636 | . . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘𝑖) = ∅) |
13 | 11, 12 | eqtr4d 2175 | . . . 4 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ((ω × {∅})‘𝑖)) |
14 | eqimss 3151 | . . . 4 ⊢ (((ω × {∅})‘suc 𝑖) = ((ω × {∅})‘𝑖) → ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)) | |
15 | 13, 14 | syl 14 | . . 3 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)) |
16 | 15 | rgen 2485 | . 2 ⊢ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖) |
17 | fveq1 5420 | . . . . 5 ⊢ (𝑓 = (ω × {∅}) → (𝑓‘suc 𝑖) = ((ω × {∅})‘suc 𝑖)) | |
18 | fveq1 5420 | . . . . 5 ⊢ (𝑓 = (ω × {∅}) → (𝑓‘𝑖) = ((ω × {∅})‘𝑖)) | |
19 | 17, 18 | sseq12d 3128 | . . . 4 ⊢ (𝑓 = (ω × {∅}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))) |
20 | 19 | ralbidv 2437 | . . 3 ⊢ (𝑓 = (ω × {∅}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))) |
21 | df-nninf 7007 | . . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
22 | 20, 21 | elrab2 2843 | . 2 ⊢ ((ω × {∅}) ∈ ℕ∞ ↔ ((ω × {∅}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))) |
23 | 7, 16, 22 | mpbir2an 926 | 1 ⊢ (ω × {∅}) ∈ ℕ∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 ∀wral 2416 ⊆ wss 3071 ∅c0 3363 {csn 3527 suc csuc 4287 ωcom 4504 × cxp 4537 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 2oc2o 6307 ↑𝑚 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-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-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-rab 2425 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-id 4215 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-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 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: exmidsbthrlem 13217 |
Copyright terms: Public domain | W3C validator |