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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 7796 | . 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} | |
2 | velsn 4141 | . . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
3 | fn0 5924 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅) | |
4 | ral0 4028 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴 | |
5 | 4 | biantru 525 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)) |
6 | 2, 3, 5 | 3bitr2i 287 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)) |
7 | 6 | abbi2i 2725 | . 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} |
8 | 1, 7 | eqtr4i 2635 | 1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∅c0 3874 {csn 4125 Fn wfn 5799 ‘cfv 5804 Xcixp 7794 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 df-fn 5807 df-ixp 7795 |
This theorem is referenced by: 0elixp 7825 ptcmpfi 21426 finixpnum 32564 ioorrnopn 39201 ioorrnopnxr 39203 hoicvr 39438 ovnhoi 39493 ovnlecvr2 39500 hoiqssbl 39515 hoimbl 39521 iunhoiioo 39567 |
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