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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 5312 | . . . . 5 ⊢ ∅ ∈ V | |
2 | vex 3466 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 5480 | . . . 4 ⊢ 〈∅, 𝑦〉 ≠ ∅ |
4 | 3 | nesymi 2988 | . . 3 ⊢ ¬ ∅ = 〈∅, 𝑦〉 |
5 | peano1 7900 | . . . . 5 ⊢ ∅ ∈ ω | |
6 | df-finxp 37091 | . . . . . 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 | eqabri 2870 | . . . . 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 5470 | . . . . . 6 ⊢ 〈∅, 𝑦〉 ∈ V | |
10 | 9 | rdg0 8451 | . . . . 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 2739 | . . . 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 274 | . . 3 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = 〈∅, 𝑦〉) |
13 | 4, 12 | mtbir 322 | . 2 ⊢ ¬ 𝑦 ∈ (𝑈↑↑∅) |
14 | 13 | nel0 4353 | 1 ⊢ (𝑈↑↑∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 ifcif 4533 〈cop 4639 ∪ cuni 4913 × cxp 5680 ‘cfv 6554 ∈ cmpo 7426 ωcom 7876 1st c1st 8001 reccrdg 8439 1oc1o 8489 ↑↑cfinxp 37090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-finxp 37091 |
This theorem is referenced by: finxp00 37109 |
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