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Mirrors > Home > MPE Home > Th. List > xpsneng | Structured version Visualization version GIF version |
Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.) |
Ref | Expression |
---|---|
xpsneng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1 5565 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 × {𝑦}) = (𝐴 × {𝑦})) | |
2 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | 1, 2 | breq12d 5066 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 × {𝑦}) ≈ 𝑥 ↔ (𝐴 × {𝑦}) ≈ 𝐴)) |
4 | sneq 4551 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵}) | |
5 | 4 | xpeq2d 5581 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 × {𝑦}) = (𝐴 × {𝐵})) |
6 | 5 | breq1d 5063 | . 2 ⊢ (𝑦 = 𝐵 → ((𝐴 × {𝑦}) ≈ 𝐴 ↔ (𝐴 × {𝐵}) ≈ 𝐴)) |
7 | vex 3412 | . . 3 ⊢ 𝑥 ∈ V | |
8 | vex 3412 | . . 3 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | xpsnen 8729 | . 2 ⊢ (𝑥 × {𝑦}) ≈ 𝑥 |
10 | 3, 6, 9 | vtocl2g 3486 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {csn 4541 class class class wbr 5053 × cxp 5549 ≈ cen 8623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-int 4860 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-en 8627 |
This theorem is referenced by: xp1en 8731 xpsnen2g 8738 xpdom3 8743 disjen 8803 unxpdom2 8886 sucxpdom 8887 gchxpidm 10283 frlmiscvec 20811 |
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