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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 7962 | . . . 4 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5159 | . . 3 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ V) |
3 | 1 | brrelex2i 5159 | . . 3 ⊢ (1𝑜 ≺ 𝐵 → 𝐵 ∈ V) |
4 | 2, 3 | anim12i 590 | . 2 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | breq2 4657 | . . . . 5 ⊢ (𝑥 = 𝐴 → (1𝑜 ≺ 𝑥 ↔ 1𝑜 ≺ 𝐴)) | |
6 | 5 | anbi1d 741 | . . . 4 ⊢ (𝑥 = 𝐴 → ((1𝑜 ≺ 𝑥 ∧ 1𝑜 ≺ 𝑦) ↔ (1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦))) |
7 | uneq1 3760 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
8 | xpeq1 5128 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦)) | |
9 | 7, 8 | breq12d 4666 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦))) |
10 | 6, 9 | imbi12d 334 | . . 3 ⊢ (𝑥 = 𝐴 → (((1𝑜 ≺ 𝑥 ∧ 1𝑜 ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)))) |
11 | breq2 4657 | . . . . 5 ⊢ (𝑦 = 𝐵 → (1𝑜 ≺ 𝑦 ↔ 1𝑜 ≺ 𝐵)) | |
12 | 11 | anbi2d 740 | . . . 4 ⊢ (𝑦 = 𝐵 → ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦) ↔ (1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵))) |
13 | uneq2 3761 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
14 | xpeq2 5129 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵)) | |
15 | 13, 14 | breq12d 4666 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))) |
16 | 12, 15 | imbi12d 334 | . . 3 ⊢ (𝑦 = 𝐵 → (((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)))) |
17 | eqid 2622 | . . . 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 2622 | . . . 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 8166 | . . 3 ⊢ ((1𝑜 ≺ 𝑥 ∧ 1𝑜 ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) |
20 | 10, 16, 19 | vtocl2g 3270 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))) |
21 | 4, 20 | mpcom 38 | 1 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ifcif 4086 〈cop 4183 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 1𝑜c1o 7553 ≼ cdom 7953 ≺ csdm 7954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 |
This theorem is referenced by: unxpdom2 8168 sucxpdom 8169 cdaxpdom 9011 |
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