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Mirrors > Home > ILE Home > Th. List > xp1en | 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 6427 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | xpeq2i 4646 | . 2 ⊢ (𝐴 × 1o) = (𝐴 × {∅}) |
3 | 0ex 4129 | . . 3 ⊢ ∅ ∈ V | |
4 | xpsneng 6819 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴) | |
5 | 3, 4 | mpan2 425 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ≈ 𝐴) |
6 | 2, 5 | eqbrtrid 4037 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 1o) ≈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 ∅c0 3422 {csn 3592 class class class wbr 4002 × cxp 4623 1oc1o 6407 ≈ cen 6735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-suc 4370 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-1o 6414 df-en 6738 |
This theorem is referenced by: (None) |
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