Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ixp0x | GIF version |
Description: An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.) |
Ref | Expression |
---|---|
ixp0x | ⊢ X𝑥 ∈ ∅ 𝐴 = {∅} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfixp 6657 | . 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} | |
2 | velsn 3587 | . . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
3 | fn0 5301 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅) | |
4 | ral0 3505 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴 | |
5 | 4 | biantru 300 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)) |
6 | 2, 3, 5 | 3bitr2i 207 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)) |
7 | 6 | abbi2i 2279 | . 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} |
8 | 1, 7 | eqtr4i 2188 | 1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∈ wcel 2135 {cab 2150 ∀wral 2442 ∅c0 3404 {csn 3570 Fn wfn 5177 ‘cfv 5182 Xcixp 6655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-fun 5184 df-fn 5185 df-ixp 6656 |
This theorem is referenced by: 0elixp 6686 |
Copyright terms: Public domain | W3C validator |