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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 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 2 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 3 | 1, 2 | opnzi 5479 | . . . 4 ⊢ 〈∅, 𝑦〉 ≠ ∅ |
| 4 | 3 | nesymi 2998 | . . 3 ⊢ ¬ ∅ = 〈∅, 𝑦〉 |
| 5 | peano1 7910 | . . . . 5 ⊢ ∅ ∈ ω | |
| 6 | df-finxp 37385 | . . . . . 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 2885 | . . . . 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 709 | . . . 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 5469 | . . . . . 6 ⊢ 〈∅, 𝑦〉 ∈ V | |
| 10 | 9 | rdg0 8461 | . . . . 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 2750 | . . . 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 275 | . . 3 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = 〈∅, 𝑦〉) |
| 13 | 4, 12 | mtbir 323 | . 2 ⊢ ¬ 𝑦 ∈ (𝑈↑↑∅) |
| 14 | 13 | nel0 4354 | 1 ⊢ (𝑈↑↑∅) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ifcif 4525 〈cop 4632 ∪ cuni 4907 × cxp 5683 ‘cfv 6561 ∈ cmpo 7433 ωcom 7887 1st c1st 8012 reccrdg 8449 1oc1o 8499 ↑↑cfinxp 37384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-finxp 37385 |
| This theorem is referenced by: finxp00 37403 |
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