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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 7839 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | xpeq2i 5369 | . 2 ⊢ (𝐴 × 1o) = (𝐴 × {∅}) |
3 | 0ex 5014 | . . 3 ⊢ ∅ ∈ V | |
4 | xpsneng 8314 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴) | |
5 | 3, 4 | mpan2 684 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ≈ 𝐴) |
6 | 2, 5 | syl5eqbr 4908 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 1o) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 Vcvv 3414 ∅c0 4144 {csn 4397 class class class wbr 4873 × cxp 5340 1oc1o 7819 ≈ cen 8219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-int 4698 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-suc 5969 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-1o 7826 df-en 8223 |
This theorem is referenced by: (None) |
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