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Mirrors > Home > MPE Home > Th. List > unxpdom | Structured version Visualization version GIF version |
Description: Cartesian product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
unxpdom | ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 7848 | . . . 4 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5083 | . . 3 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ V) |
3 | 1 | brrelex2i 5083 | . . 3 ⊢ (1𝑜 ≺ 𝐵 → 𝐵 ∈ V) |
4 | 2, 3 | anim12i 588 | . 2 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | breq2 4587 | . . . . 5 ⊢ (𝑥 = 𝐴 → (1𝑜 ≺ 𝑥 ↔ 1𝑜 ≺ 𝐴)) | |
6 | 5 | anbi1d 737 | . . . 4 ⊢ (𝑥 = 𝐴 → ((1𝑜 ≺ 𝑥 ∧ 1𝑜 ≺ 𝑦) ↔ (1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦))) |
7 | uneq1 3722 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
8 | xpeq1 5052 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦)) | |
9 | 7, 8 | breq12d 4596 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦))) |
10 | 6, 9 | imbi12d 333 | . . 3 ⊢ (𝑥 = 𝐴 → (((1𝑜 ≺ 𝑥 ∧ 1𝑜 ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)))) |
11 | breq2 4587 | . . . . 5 ⊢ (𝑦 = 𝐵 → (1𝑜 ≺ 𝑦 ↔ 1𝑜 ≺ 𝐵)) | |
12 | 11 | anbi2d 736 | . . . 4 ⊢ (𝑦 = 𝐵 → ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦) ↔ (1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵))) |
13 | uneq2 3723 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
14 | xpeq2 5053 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵)) | |
15 | 13, 14 | breq12d 4596 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))) |
16 | 12, 15 | imbi12d 333 | . . 3 ⊢ (𝑦 = 𝐵 → (((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)))) |
17 | eqid 2610 | . . . 4 ⊢ (𝑧 ∈ (𝑥 ∪ 𝑦) ↦ if(𝑧 ∈ 𝑥, 〈𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)〉, 〈if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧〉)) = (𝑧 ∈ (𝑥 ∪ 𝑦) ↦ if(𝑧 ∈ 𝑥, 〈𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)〉, 〈if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧〉)) | |
18 | eqid 2610 | . . . 4 ⊢ if(𝑧 ∈ 𝑥, 〈𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)〉, 〈if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧〉) = if(𝑧 ∈ 𝑥, 〈𝑧, if(𝑧 = 𝑣, 𝑤, 𝑡)〉, 〈if(𝑧 = 𝑤, 𝑢, 𝑣), 𝑧〉) | |
19 | 17, 18 | unxpdomlem3 8051 | . . 3 ⊢ ((1𝑜 ≺ 𝑥 ∧ 1𝑜 ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) |
20 | 10, 16, 19 | vtocl2g 3243 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))) |
21 | 4, 20 | mpcom 37 | 1 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ifcif 4036 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 1𝑜c1o 7440 ≼ cdom 7839 ≺ csdm 7840 |
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-3or 1032 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-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-2o 7448 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: unxpdom2 8053 sucxpdom 8054 cdaxpdom 8894 |
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