![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xpsnen2g | Structured version Visualization version GIF version |
Description: A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Ref | Expression |
---|---|
xpsnen2g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × 𝐵) ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5393 | . . 3 ⊢ {𝐴} ∈ V | |
2 | xpcomeng 9015 | . . 3 ⊢ (({𝐴} ∈ V ∧ 𝐵 ∈ 𝑊) → ({𝐴} × 𝐵) ≈ (𝐵 × {𝐴})) | |
3 | 1, 2 | mpan 689 | . 2 ⊢ (𝐵 ∈ 𝑊 → ({𝐴} × 𝐵) ≈ (𝐵 × {𝐴})) |
4 | xpsneng 9007 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × {𝐴}) ≈ 𝐵) | |
5 | 4 | ancoms 460 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝐴}) ≈ 𝐵) |
6 | entr 8953 | . 2 ⊢ ((({𝐴} × 𝐵) ≈ (𝐵 × {𝐴}) ∧ (𝐵 × {𝐴}) ≈ 𝐵) → ({𝐴} × 𝐵) ≈ 𝐵) | |
7 | 3, 5, 6 | syl2an2 685 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × 𝐵) ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3448 {csn 4591 class class class wbr 5110 × cxp 5636 ≈ cen 8887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-1st 7926 df-2nd 7927 df-er 8655 df-en 8891 |
This theorem is referenced by: unxpwdom2 9531 undjudom 10110 endjudisj 10111 djuen 10112 dju1dif 10115 dju1p1e2 10116 djucomen 10120 djuassen 10121 xpdjuen 10122 mapdjuen 10123 djuxpdom 10128 djufi 10129 djuinf 10131 infdju1 10132 pwdjudom 10159 ackbij1lem8 10170 isfin4p1 10258 pwdjundom 10610 lgsquadlem1 26744 lgsquadlem2 26745 |
Copyright terms: Public domain | W3C validator |