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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 8836 | . 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} | |
2 | velsn 4602 | . . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
3 | fn0 6632 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅) | |
4 | ral0 4470 | . . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴 | |
5 | 4 | biantru 530 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)) |
6 | 2, 3, 5 | 3bitr2i 298 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)) |
7 | 6 | abbi2i 2873 | . 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} |
8 | 1, 7 | eqtr4i 2767 | 1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2713 ∀wral 3064 ∅c0 4282 {csn 4586 Fn wfn 6491 ‘cfv 6496 Xcixp 8834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2538 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-fun 6498 df-fn 6499 df-ixp 8835 |
This theorem is referenced by: 0elixp 8866 ptcmpfi 23162 finixpnum 36053 ioorrnopn 44517 ioorrnopnxr 44519 hoicvr 44760 ovnhoi 44815 ovnlecvr2 44822 hoiqssbl 44837 hoimbl 44843 iunhoiioo 44888 |
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