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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 8844 | . 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} | |
2 | velsn 4607 | . . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
3 | fn0 6637 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅) | |
4 | ral0 4475 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴 | |
5 | 4 | biantru 531 | . . . 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 | abbi2i 2874 | . 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} |
8 | 1, 7 | eqtr4i 2768 | 1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2714 ∀wral 3065 ∅c0 4287 {csn 4591 Fn wfn 6496 ‘cfv 6501 Xcixp 8842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-fun 6503 df-fn 6504 df-ixp 8843 |
This theorem is referenced by: 0elixp 8874 ptcmpfi 23180 finixpnum 36092 ioorrnopn 44620 ioorrnopnxr 44622 hoicvr 44863 ovnhoi 44918 ovnlecvr2 44925 hoiqssbl 44940 hoimbl 44946 iunhoiioo 44991 |
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