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Mirrors > Home > MPE Home > Th. List > cdaxpdom | Structured version Visualization version GIF version |
Description: Cartesian product dominates disjoint union for sets with cardinality greater than 1. Similar to Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
cdaxpdom | ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsdom 7848 | . . . . 5 ⊢ Rel ≺ | |
2 | 1 | brrelex2i 5083 | . . . 4 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ V) |
3 | 1 | brrelex2i 5083 | . . . 4 ⊢ (1𝑜 ≺ 𝐵 → 𝐵 ∈ V) |
4 | cdaval 8875 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜}))) | |
5 | 2, 3, 4 | syl2an 493 | . . 3 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜}))) |
6 | 0ex 4718 | . . . . . . 7 ⊢ ∅ ∈ V | |
7 | xpsneng 7930 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴) | |
8 | 2, 6, 7 | sylancl 693 | . . . . . 6 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 × {∅}) ≈ 𝐴) |
9 | sdomen2 7990 | . . . . . 6 ⊢ ((𝐴 × {∅}) ≈ 𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜 ≺ 𝐴)) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (1𝑜 ≺ 𝐴 → (1𝑜 ≺ (𝐴 × {∅}) ↔ 1𝑜 ≺ 𝐴)) |
11 | 10 | ibir 256 | . . . 4 ⊢ (1𝑜 ≺ 𝐴 → 1𝑜 ≺ (𝐴 × {∅})) |
12 | 1on 7454 | . . . . . . 7 ⊢ 1𝑜 ∈ On | |
13 | xpsneng 7930 | . . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 1𝑜 ∈ On) → (𝐵 × {1𝑜}) ≈ 𝐵) | |
14 | 3, 12, 13 | sylancl 693 | . . . . . 6 ⊢ (1𝑜 ≺ 𝐵 → (𝐵 × {1𝑜}) ≈ 𝐵) |
15 | sdomen2 7990 | . . . . . 6 ⊢ ((𝐵 × {1𝑜}) ≈ 𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜 ≺ 𝐵)) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ (1𝑜 ≺ 𝐵 → (1𝑜 ≺ (𝐵 × {1𝑜}) ↔ 1𝑜 ≺ 𝐵)) |
17 | 16 | ibir 256 | . . . 4 ⊢ (1𝑜 ≺ 𝐵 → 1𝑜 ≺ (𝐵 × {1𝑜})) |
18 | unxpdom 8052 | . . . 4 ⊢ ((1𝑜 ≺ (𝐴 × {∅}) ∧ 1𝑜 ≺ (𝐵 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜}))) | |
19 | 11, 17, 18 | syl2an 493 | . . 3 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜}))) |
20 | 5, 19 | eqbrtrd 4605 | . 2 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜}))) |
21 | xpen 8008 | . . 3 ⊢ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵)) | |
22 | 8, 14, 21 | syl2an 493 | . 2 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵)) |
23 | domentr 7901 | . 2 ⊢ (((𝐴 +𝑐 𝐵) ≼ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ∧ ((𝐴 × {∅}) × (𝐵 × {1𝑜})) ≈ (𝐴 × 𝐵)) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵)) | |
24 | 20, 22, 23 | syl2anc 691 | 1 ⊢ ((1𝑜 ≺ 𝐴 ∧ 1𝑜 ≺ 𝐵) → (𝐴 +𝑐 𝐵) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∅c0 3874 {csn 4125 class class class wbr 4583 × cxp 5036 Oncon0 5640 (class class class)co 6549 1𝑜c1o 7440 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 +𝑐 ccda 8872 |
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-csb 3500 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-int 4411 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-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-1o 7447 df-2o 7448 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-cda 8873 |
This theorem is referenced by: canthp1lem1 9353 |
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