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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 5437 | . . 3 ⊢ {𝐴} ∈ V | |
2 | xpcomeng 9097 | . . 3 ⊢ (({𝐴} ∈ V ∧ 𝐵 ∈ 𝑊) → ({𝐴} × 𝐵) ≈ (𝐵 × {𝐴})) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝐵 ∈ 𝑊 → ({𝐴} × 𝐵) ≈ (𝐵 × {𝐴})) |
4 | xpsneng 9089 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × {𝐴}) ≈ 𝐵) | |
5 | 4 | ancoms 457 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝐴}) ≈ 𝐵) |
6 | entr 9035 | . 2 ⊢ ((({𝐴} × 𝐵) ≈ (𝐵 × {𝐴}) ∧ (𝐵 × {𝐴}) ≈ 𝐵) → ({𝐴} × 𝐵) ≈ 𝐵) | |
7 | 3, 5, 6 | syl2an2 684 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × 𝐵) ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3473 {csn 4632 class class class wbr 5152 × cxp 5680 ≈ cen 8969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-1st 8001 df-2nd 8002 df-er 8733 df-en 8973 |
This theorem is referenced by: unxpwdom2 9621 undjudom 10200 endjudisj 10201 djuen 10202 dju1dif 10205 dju1p1e2 10206 djucomen 10210 djuassen 10211 xpdjuen 10212 mapdjuen 10213 djuxpdom 10218 djufi 10219 djuinf 10221 infdju1 10222 pwdjudom 10249 ackbij1lem8 10260 isfin4p1 10348 pwdjundom 10700 lgsquadlem1 27341 lgsquadlem2 27342 |
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