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Mirrors > Home > MPE Home > Th. List > ixp0x | Structured version Visualization version 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 8917 | . 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} | |
2 | velsn 4645 | . . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
3 | fn0 6686 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅) | |
4 | ral0 4513 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴 | |
5 | 4 | biantru 529 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)) |
6 | 2, 3, 5 | 3bitr2i 299 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)) |
7 | 6 | eqabi 2865 | . 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} |
8 | 1, 7 | eqtr4i 2759 | 1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 {cab 2705 ∀wral 3058 ∅c0 4323 {csn 4629 Fn wfn 6543 ‘cfv 6548 Xcixp 8915 |
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-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-fun 6550 df-fn 6551 df-ixp 8916 |
This theorem is referenced by: 0elixp 8947 ptcmpfi 23716 finixpnum 37078 ioorrnopn 45693 ioorrnopnxr 45695 hoicvr 45936 ovnhoi 45991 ovnlecvr2 45998 hoiqssbl 46013 hoimbl 46019 iunhoiioo 46064 |
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