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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 7007 | . 2 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
2 | 2onn 6417 | . . . . . 6 ⊢ 2o ∈ ω | |
3 | 2 | elexi 2698 | . . . . 5 ⊢ 2o ∈ V |
4 | omex 4507 | . . . . 5 ⊢ ω ∈ V | |
5 | 3, 4 | mapval 6554 | . . . 4 ⊢ (2o ↑𝑚 ω) = {𝑔 ∣ 𝑔:ω⟶2o} |
6 | mapex 6548 | . . . . 5 ⊢ ((ω ∈ V ∧ 2o ∈ V) → {𝑔 ∣ 𝑔:ω⟶2o} ∈ V) | |
7 | 4, 3, 6 | mp2an 422 | . . . 4 ⊢ {𝑔 ∣ 𝑔:ω⟶2o} ∈ V |
8 | 5, 7 | eqeltri 2212 | . . 3 ⊢ (2o ↑𝑚 ω) ∈ V |
9 | 8 | rabex 4072 | . 2 ⊢ {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} ∈ V |
10 | 1, 9 | eqeltri 2212 | 1 ⊢ ℕ∞ ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 {cab 2125 ∀wral 2416 {crab 2420 Vcvv 2686 ⊆ wss 3071 suc csuc 4287 ωcom 4504 ⟶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-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: nninfomnilem 13214 nninffeq 13216 exmidsbthrlem 13217 |
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