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Mathbox for Jim Kingdon |
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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 6346 | . . . 4 ⊢ ∅ ∈ 2o | |
2 | 1 | fconst6 5330 | . . 3 ⊢ (ω × {∅}):ω⟶2o |
3 | 2onn 6425 | . . . . 5 ⊢ 2o ∈ ω | |
4 | 3 | elexi 2701 | . . . 4 ⊢ 2o ∈ V |
5 | omex 4515 | . . . 4 ⊢ ω ∈ V | |
6 | 4, 5 | elmap 6579 | . . 3 ⊢ ((ω × {∅}) ∈ (2o ↑𝑚 ω) ↔ (ω × {∅}):ω⟶2o) |
7 | 2, 6 | mpbir 145 | . 2 ⊢ (ω × {∅}) ∈ (2o ↑𝑚 ω) |
8 | peano2 4517 | . . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω) | |
9 | 0ex 4063 | . . . . . . 7 ⊢ ∅ ∈ V | |
10 | 9 | fvconst2 5644 | . . . . . 6 ⊢ (suc 𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ∅) |
11 | 8, 10 | syl 14 | . . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ∅) |
12 | 9 | fvconst2 5644 | . . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘𝑖) = ∅) |
13 | 11, 12 | eqtr4d 2176 | . . . 4 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ((ω × {∅})‘𝑖)) |
14 | eqimss 3156 | . . . 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 2488 | . 2 ⊢ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖) |
17 | fveq1 5428 | . . . . 5 ⊢ (𝑓 = (ω × {∅}) → (𝑓‘suc 𝑖) = ((ω × {∅})‘suc 𝑖)) | |
18 | fveq1 5428 | . . . . 5 ⊢ (𝑓 = (ω × {∅}) → (𝑓‘𝑖) = ((ω × {∅})‘𝑖)) | |
19 | 17, 18 | sseq12d 3133 | . . . 4 ⊢ (𝑓 = (ω × {∅}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))) |
20 | 19 | ralbidv 2438 | . . 3 ⊢ (𝑓 = (ω × {∅}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))) |
21 | df-nninf 7015 | . . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
22 | 20, 21 | elrab2 2847 | . 2 ⊢ ((ω × {∅}) ∈ ℕ∞ ↔ ((ω × {∅}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))) |
23 | 7, 16, 22 | mpbir2an 927 | 1 ⊢ (ω × {∅}) ∈ ℕ∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 ∀wral 2417 ⊆ wss 3076 ∅c0 3368 {csn 3532 suc csuc 4295 ωcom 4512 × cxp 4545 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 2oc2o 6315 ↑𝑚 cmap 6550 ℕ∞xnninf 7013 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1o 6321 df-2o 6322 df-map 6552 df-nninf 7015 |
This theorem is referenced by: exmidsbthrlem 13392 |
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