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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 5313 | . . . . 5 ⊢ ∅ ∈ V | |
2 | vex 3482 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 5485 | . . . 4 ⊢ 〈∅, 𝑦〉 ≠ ∅ |
4 | 3 | nesymi 2996 | . . 3 ⊢ ¬ ∅ = 〈∅, 𝑦〉 |
5 | peano1 7911 | . . . . 5 ⊢ ∅ ∈ ω | |
6 | df-finxp 37367 | . . . . . 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 2883 | . . . . 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 5475 | . . . . . 6 ⊢ 〈∅, 𝑦〉 ∈ V | |
10 | 9 | rdg0 8460 | . . . . 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 2748 | . . . 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 4360 | 1 ⊢ (𝑈↑↑∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ifcif 4531 〈cop 4637 ∪ cuni 4912 × cxp 5687 ‘cfv 6563 ∈ cmpo 7433 ωcom 7887 1st c1st 8011 reccrdg 8448 1oc1o 8498 ↑↑cfinxp 37366 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-finxp 37367 |
This theorem is referenced by: finxp00 37385 |
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