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Mirrors > Home > ILE Home > Th. List > infnninfOLD | GIF version |
Description: Obsolete version of infnninf 7185 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 6497 | . . . 4 ⊢ 1o ∈ 2o | |
2 | 1 | fconst6 5454 | . . 3 ⊢ (ω × {1o}):ω⟶2o |
3 | 2onn 6576 | . . . . 5 ⊢ 2o ∈ ω | |
4 | 3 | elexi 2772 | . . . 4 ⊢ 2o ∈ V |
5 | omex 4626 | . . . 4 ⊢ ω ∈ V | |
6 | 4, 5 | elmap 6733 | . . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o) |
7 | 2, 6 | mpbir 146 | . 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω) |
8 | peano2 4628 | . . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω) | |
9 | 1oex 6479 | . . . . . . 7 ⊢ 1o ∈ V | |
10 | 9 | fvconst2 5775 | . . . . . 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 5775 | . . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o) |
13 | 11, 12 | eqtr4d 2229 | . . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖)) |
14 | eqimss 3234 | . . . 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 2547 | . 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖) |
17 | fveq1 5554 | . . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖)) | |
18 | fveq1 5554 | . . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖)) | |
19 | 17, 18 | sseq12d 3211 | . . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
20 | 19 | ralbidv 2494 | . . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
21 | df-nninf 7181 | . . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
22 | 20, 21 | elrab2 2920 | . 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
23 | 7, 16, 22 | mpbir2an 944 | 1 ⊢ (ω × {1o}) ∈ ℕ∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2164 ∀wral 2472 ⊆ wss 3154 {csn 3619 suc csuc 4397 ωcom 4623 × cxp 4658 ⟶wf 5251 ‘cfv 5255 (class class class)co 5919 1oc1o 6464 2oc2o 6465 ↑𝑚 cmap 6704 ℕ∞xnninf 7180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-iord 4398 df-on 4400 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1o 6471 df-2o 6472 df-map 6706 df-nninf 7181 |
This theorem is referenced by: fxnn0nninf 10513 |
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