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Mirrors > Home > ILE Home > Th. List > infnninfOLD | GIF version |
Description: Obsolete version of infnninf 7100 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 6421 | . . . 4 ⊢ 1o ∈ 2o | |
2 | 1 | fconst6 5397 | . . 3 ⊢ (ω × {1o}):ω⟶2o |
3 | 2onn 6500 | . . . . 5 ⊢ 2o ∈ ω | |
4 | 3 | elexi 2742 | . . . 4 ⊢ 2o ∈ V |
5 | omex 4577 | . . . 4 ⊢ ω ∈ V | |
6 | 4, 5 | elmap 6655 | . . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o) |
7 | 2, 6 | mpbir 145 | . 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω) |
8 | peano2 4579 | . . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω) | |
9 | 1oex 6403 | . . . . . . 7 ⊢ 1o ∈ V | |
10 | 9 | fvconst2 5712 | . . . . . 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 5712 | . . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o) |
13 | 11, 12 | eqtr4d 2206 | . . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖)) |
14 | eqimss 3201 | . . . 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 2523 | . 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖) |
17 | fveq1 5495 | . . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖)) | |
18 | fveq1 5495 | . . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖)) | |
19 | 17, 18 | sseq12d 3178 | . . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
20 | 19 | ralbidv 2470 | . . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
21 | df-nninf 7097 | . . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
22 | 20, 21 | elrab2 2889 | . 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))) |
23 | 7, 16, 22 | mpbir2an 937 | 1 ⊢ (ω × {1o}) ∈ ℕ∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 ∀wral 2448 ⊆ wss 3121 {csn 3583 suc csuc 4350 ωcom 4574 × cxp 4609 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 1oc1o 6388 2oc2o 6389 ↑𝑚 cmap 6626 ℕ∞xnninf 7096 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1o 6395 df-2o 6396 df-map 6628 df-nninf 7097 |
This theorem is referenced by: fxnn0nninf 10394 |
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