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Mirrors > Home > ILE Home > Th. List > infnninfOLD | GIF version |
Description: Obsolete version of infnninf 7069 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
infnninfOLD | ⊢ (ω × {1o}) ∈ ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2o 6391 | . . . 4 ⊢ 1o ∈ 2o | |
2 | 1 | fconst6 5371 | . . 3 ⊢ (ω × {1o}):ω⟶2o |
3 | 2onn 6470 | . . . . 5 ⊢ 2o ∈ ω | |
4 | 3 | elexi 2724 | . . . 4 ⊢ 2o ∈ V |
5 | omex 4554 | . . . 4 ⊢ ω ∈ V | |
6 | 4, 5 | elmap 6624 | . . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o) |
7 | 2, 6 | mpbir 145 | . 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω) |
8 | peano2 4556 | . . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω) | |
9 | 1oex 6373 | . . . . . . 7 ⊢ 1o ∈ V | |
10 | 9 | fvconst2 5685 | . . . . . 6 ⊢ (suc 𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o) |
11 | 8, 10 | syl 14 | . . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o) |
12 | 9 | fvconst2 5685 | . . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o) |
13 | 11, 12 | eqtr4d 2193 | . . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖)) |
14 | eqimss 3182 | . . . 4 ⊢ (((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖) → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)) | |
15 | 13, 14 | syl 14 | . . 3 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)) |
16 | 15 | rgen 2510 | . 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖) |
17 | fveq1 5469 | . . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖)) | |
18 | fveq1 5469 | . . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖)) | |
19 | 17, 18 | sseq12d 3159 | . . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
20 | 19 | ralbidv 2457 | . . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
21 | df-nninf 7066 | . . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
22 | 20, 21 | elrab2 2871 | . 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
23 | 7, 16, 22 | mpbir2an 927 | 1 ⊢ (ω × {1o}) ∈ ℕ∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 ∀wral 2435 ⊆ wss 3102 {csn 3561 suc csuc 4327 ωcom 4551 × cxp 4586 ⟶wf 5168 ‘cfv 5172 (class class class)co 5826 1oc1o 6358 2oc2o 6359 ↑𝑚 cmap 6595 ℕ∞xnninf 7065 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1o 6365 df-2o 6366 df-map 6597 df-nninf 7066 |
This theorem is referenced by: fxnn0nninf 10346 |
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