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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 5052 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 × {𝑦}) = (𝐴 × {𝑦})) | |
2 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | 1, 2 | breq12d 4596 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 × {𝑦}) ≈ 𝑥 ↔ (𝐴 × {𝑦}) ≈ 𝐴)) |
4 | sneq 4135 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵}) | |
5 | 4 | xpeq2d 5063 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 × {𝑦}) = (𝐴 × {𝐵})) |
6 | 5 | breq1d 4593 | . 2 ⊢ (𝑦 = 𝐵 → ((𝐴 × {𝑦}) ≈ 𝐴 ↔ (𝐴 × {𝐵}) ≈ 𝐴)) |
7 | vex 3176 | . . 3 ⊢ 𝑥 ∈ V | |
8 | vex 3176 | . . 3 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | xpsnen 7929 | . 2 ⊢ (𝑥 × {𝑦}) ≈ 𝑥 |
10 | 3, 6, 9 | vtocl2g 3243 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 class class class wbr 4583 × cxp 5036 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-en 7842 |
This theorem is referenced by: xp1en 7931 xpsnen2g 7938 xpdom3 7943 disjen 8002 unxpdom2 8053 sucxpdom 8054 uncdadom 8876 cdaun 8877 cdaen 8878 cda1dif 8881 cdacomen 8886 cdaassen 8887 xpcdaen 8888 mapcdaen 8889 cdaxpdom 8894 cdafi 8895 cdainf 8897 infcda1 8898 pwcdadom 8921 isfin4-3 9020 pwcdandom 9368 gchxpidm 9370 frlmiscvec 20007 |
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