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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 8977 | . . . 4 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5739 | . . 3 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V) |
3 | 1 | brrelex2i 5739 | . . 3 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V) |
4 | 2, 3 | anim12i 611 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | breq2 5156 | . . . . 5 ⊢ (𝑥 = 𝐴 → (1o ≺ 𝑥 ↔ 1o ≺ 𝐴)) | |
6 | 5 | anbi1d 629 | . . . 4 ⊢ (𝑥 = 𝐴 → ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝑦))) |
7 | uneq1 4157 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
8 | xpeq1 5696 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦)) | |
9 | 7, 8 | breq12d 5165 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦) ↔ (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦))) |
10 | 6, 9 | imbi12d 343 | . . 3 ⊢ (𝑥 = 𝐴 → (((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) ↔ ((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)))) |
11 | breq2 5156 | . . . . 5 ⊢ (𝑦 = 𝐵 → (1o ≺ 𝑦 ↔ 1o ≺ 𝐵)) | |
12 | 11 | anbi2d 628 | . . . 4 ⊢ (𝑦 = 𝐵 → ((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) ↔ (1o ≺ 𝐴 ∧ 1o ≺ 𝐵))) |
13 | uneq2 4158 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
14 | xpeq2 5703 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵)) | |
15 | 13, 14 | breq12d 5165 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦) ↔ (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))) |
16 | 12, 15 | imbi12d 343 | . . 3 ⊢ (𝑦 = 𝐵 → (((1o ≺ 𝐴 ∧ 1o ≺ 𝑦) → (𝐴 ∪ 𝑦) ≼ (𝐴 × 𝑦)) ↔ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)))) |
17 | eqid 2728 | . . . 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 2728 | . . . 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 9283 | . . 3 ⊢ ((1o ≺ 𝑥 ∧ 1o ≺ 𝑦) → (𝑥 ∪ 𝑦) ≼ (𝑥 × 𝑦)) |
20 | 10, 16, 19 | vtocl2g 3562 | . 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 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∪ cun 3947 ifcif 4532 ⟨cop 4638 class class class wbr 5152 ↦ cmpt 5235 × cxp 5680 1oc1o 8486 ≼ cdom 8968 ≺ csdm 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 7746 |
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-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-suc 6380 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-1o 8493 df-2o 8494 df-en 8971 df-dom 8972 df-sdom 8973 |
This theorem is referenced by: unxpdom2 9285 sucxpdom 9286 djuxpdom 10216 |
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