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Mirrors > Home > ILE Home > Th. List > infnninfOLD | GIF version |
Description: Obsolete version of infnninf 7079 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 6401 | . . . 4 ⊢ 1o ∈ 2o | |
2 | 1 | fconst6 5381 | . . 3 ⊢ (ω × {1o}):ω⟶2o |
3 | 2onn 6480 | . . . . 5 ⊢ 2o ∈ ω | |
4 | 3 | elexi 2733 | . . . 4 ⊢ 2o ∈ V |
5 | omex 4564 | . . . 4 ⊢ ω ∈ V | |
6 | 4, 5 | elmap 6634 | . . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o) |
7 | 2, 6 | mpbir 145 | . 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω) |
8 | peano2 4566 | . . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω) | |
9 | 1oex 6383 | . . . . . . 7 ⊢ 1o ∈ V | |
10 | 9 | fvconst2 5695 | . . . . . 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 5695 | . . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o) |
13 | 11, 12 | eqtr4d 2200 | . . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖)) |
14 | eqimss 3191 | . . . 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 2517 | . 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖) |
17 | fveq1 5479 | . . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖)) | |
18 | fveq1 5479 | . . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖)) | |
19 | 17, 18 | sseq12d 3168 | . . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
20 | 19 | ralbidv 2464 | . . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
21 | df-nninf 7076 | . . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
22 | 20, 21 | elrab2 2880 | . 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
23 | 7, 16, 22 | mpbir2an 931 | 1 ⊢ (ω × {1o}) ∈ ℕ∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 ∀wral 2442 ⊆ wss 3111 {csn 3570 suc csuc 4337 ωcom 4561 × cxp 4596 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 1oc1o 6368 2oc2o 6369 ↑𝑚 cmap 6605 ℕ∞xnninf 7075 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1o 6375 df-2o 6376 df-map 6607 df-nninf 7076 |
This theorem is referenced by: fxnn0nninf 10363 |
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