![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xp1en | Structured version Visualization version GIF version |
Description: One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
xp1en | ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 1o) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 8493 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | xpeq2i 5705 | . 2 ⊢ (𝐴 × 1o) = (𝐴 × {∅}) |
3 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
4 | xpsneng 9080 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴) | |
5 | 3, 4 | mpan2 690 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ≈ 𝐴) |
6 | 2, 5 | eqbrtrid 5183 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 1o) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3471 ∅c0 4323 {csn 4629 class class class wbr 5148 × cxp 5676 1oc1o 8479 ≈ cen 8960 |
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-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 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-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-suc 6375 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-1o 8486 df-en 8964 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |