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Mirrors > Home > ILE Home > Th. List > Mathboxes > nninfex | GIF version |
Description: ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.) |
Ref | Expression |
---|---|
nninfex | ⊢ ℕ∞ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nninf 7015 | . 2 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
2 | 2onn 6425 | . . . . . 6 ⊢ 2o ∈ ω | |
3 | 2 | elexi 2701 | . . . . 5 ⊢ 2o ∈ V |
4 | omex 4515 | . . . . 5 ⊢ ω ∈ V | |
5 | 3, 4 | mapval 6562 | . . . 4 ⊢ (2o ↑𝑚 ω) = {𝑔 ∣ 𝑔:ω⟶2o} |
6 | mapex 6556 | . . . . 5 ⊢ ((ω ∈ V ∧ 2o ∈ V) → {𝑔 ∣ 𝑔:ω⟶2o} ∈ V) | |
7 | 4, 3, 6 | mp2an 423 | . . . 4 ⊢ {𝑔 ∣ 𝑔:ω⟶2o} ∈ V |
8 | 5, 7 | eqeltri 2213 | . . 3 ⊢ (2o ↑𝑚 ω) ∈ V |
9 | 8 | rabex 4080 | . 2 ⊢ {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} ∈ V |
10 | 1, 9 | eqeltri 2213 | 1 ⊢ ℕ∞ ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 {cab 2126 ∀wral 2417 {crab 2421 Vcvv 2689 ⊆ wss 3076 suc csuc 4295 ωcom 4512 ⟶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-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: nninfomnilem 13389 nninffeq 13391 exmidsbthrlem 13392 |
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