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Mirrors > Home > MPE Home > Th. List > Mathboxes > finxp0 | Structured version Visualization version GIF version |
Description: The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.) |
Ref | Expression |
---|---|
finxp0 | ⊢ (𝑈↑↑∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5204 | . . . . 5 ⊢ ∅ ∈ V | |
2 | vex 3498 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 5359 | . . . 4 ⊢ 〈∅, 𝑦〉 ≠ ∅ |
4 | 3 | nesymi 3073 | . . 3 ⊢ ¬ ∅ = 〈∅, 𝑦〉 |
5 | peano1 7595 | . . . . 5 ⊢ ∅ ∈ ω | |
6 | df-finxp 34659 | . . . . . 6 ⊢ (𝑈↑↑∅) = {𝑦 ∣ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈∅, 𝑦〉)‘∅))} | |
7 | 6 | abeq2i 2948 | . . . . 5 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ (∅ ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈∅, 𝑦〉)‘∅))) |
8 | 5, 7 | mpbiran 707 | . . . 4 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈∅, 𝑦〉)‘∅)) |
9 | opex 5349 | . . . . . 6 ⊢ 〈∅, 𝑦〉 ∈ V | |
10 | 9 | rdg0 8051 | . . . . 5 ⊢ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈∅, 𝑦〉)‘∅) = 〈∅, 𝑦〉 |
11 | 10 | eqeq2i 2834 | . . . 4 ⊢ (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈∅, 𝑦〉)‘∅) ↔ ∅ = 〈∅, 𝑦〉) |
12 | 8, 11 | bitri 277 | . . 3 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = 〈∅, 𝑦〉) |
13 | 4, 12 | mtbir 325 | . 2 ⊢ ¬ 𝑦 ∈ (𝑈↑↑∅) |
14 | 13 | nel0 4311 | 1 ⊢ (𝑈↑↑∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∅c0 4291 ifcif 4467 〈cop 4567 ∪ cuni 4832 × cxp 5548 ‘cfv 6350 ∈ cmpo 7152 ωcom 7574 1st c1st 7681 reccrdg 8039 1oc1o 8089 ↑↑cfinxp 34658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-finxp 34659 |
This theorem is referenced by: finxp00 34677 |
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