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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 4718 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 4869 | . . . . 5 ⊢ 〈∅, 𝑦〉 ≠ ∅ |
4 | 3 | nesymi 2839 | . . . 4 ⊢ ¬ ∅ = 〈∅, 𝑦〉 |
5 | peano1 6977 | . . . . . 6 ⊢ ∅ ∈ ω | |
6 | df-finxp 32397 | . . . . . . 7 ⊢ (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 2722 | . . . . . 6 ⊢ (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 955 | . . . . 5 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈∅, 𝑦〉)‘∅)) |
9 | opex 4859 | . . . . . . 7 ⊢ 〈∅, 𝑦〉 ∈ V | |
10 | 9 | rdg0 7404 | . . . . . 6 ⊢ (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈∅, 𝑦〉)‘∅) = 〈∅, 𝑦〉 |
11 | 10 | eqeq2i 2622 | . . . . 5 ⊢ (∅ = (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 263 | . . . 4 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ ∅ = 〈∅, 𝑦〉) |
13 | 4, 12 | mtbir 312 | . . 3 ⊢ ¬ 𝑦 ∈ (𝑈↑↑∅) |
14 | noel 3878 | . . 3 ⊢ ¬ 𝑦 ∈ ∅ | |
15 | 13, 14 | 2false 364 | . 2 ⊢ (𝑦 ∈ (𝑈↑↑∅) ↔ 𝑦 ∈ ∅) |
16 | 15 | eqriv 2607 | 1 ⊢ (𝑈↑↑∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ifcif 4036 〈cop 4131 ∪ cuni 4372 × cxp 5036 ‘cfv 5804 ↦ cmpt2 6551 ωcom 6957 1st c1st 7057 reccrdg 7392 1𝑜c1o 7440 ↑↑cfinxp 32396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-finxp 32397 |
This theorem is referenced by: finxp00 32415 |
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