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Mirrors > Home > MPE Home > Th. List > djuxpdom | 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 |
---|---|
djuxpdom | ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9893 | . . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 0ex 5298 | . . . . . . 7 ⊢ ∅ ∈ V | |
3 | relsdom 8943 | . . . . . . . 8 ⊢ Rel ≺ | |
4 | 3 | brrelex2i 5724 | . . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V) |
5 | xpsnen2g 9062 | . . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
6 | 2, 4, 5 | sylancr 586 | . . . . . 6 ⊢ (1o ≺ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
7 | sdomen2 9119 | . . . . . 6 ⊢ (({∅} × 𝐴) ≈ 𝐴 → (1o ≺ ({∅} × 𝐴) ↔ 1o ≺ 𝐴)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (1o ≺ 𝐴 → (1o ≺ ({∅} × 𝐴) ↔ 1o ≺ 𝐴)) |
9 | 8 | ibir 268 | . . . 4 ⊢ (1o ≺ 𝐴 → 1o ≺ ({∅} × 𝐴)) |
10 | 1on 8474 | . . . . . . 7 ⊢ 1o ∈ On | |
11 | 3 | brrelex2i 5724 | . . . . . . 7 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V) |
12 | xpsnen2g 9062 | . . . . . . 7 ⊢ ((1o ∈ On ∧ 𝐵 ∈ V) → ({1o} × 𝐵) ≈ 𝐵) | |
13 | 10, 11, 12 | sylancr 586 | . . . . . 6 ⊢ (1o ≺ 𝐵 → ({1o} × 𝐵) ≈ 𝐵) |
14 | sdomen2 9119 | . . . . . 6 ⊢ (({1o} × 𝐵) ≈ 𝐵 → (1o ≺ ({1o} × 𝐵) ↔ 1o ≺ 𝐵)) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (1o ≺ 𝐵 → (1o ≺ ({1o} × 𝐵) ↔ 1o ≺ 𝐵)) |
16 | 15 | ibir 268 | . . . 4 ⊢ (1o ≺ 𝐵 → 1o ≺ ({1o} × 𝐵)) |
17 | unxpdom 9250 | . . . 4 ⊢ ((1o ≺ ({∅} × 𝐴) ∧ 1o ≺ ({1o} × 𝐵)) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) | |
18 | 9, 16, 17 | syl2an 595 | . . 3 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) |
19 | 1, 18 | eqbrtrid 5174 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) |
20 | xpen 9137 | . . 3 ⊢ ((({∅} × 𝐴) ≈ 𝐴 ∧ ({1o} × 𝐵) ≈ 𝐵) → (({∅} × 𝐴) × ({1o} × 𝐵)) ≈ (𝐴 × 𝐵)) | |
21 | 6, 13, 20 | syl2an 595 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (({∅} × 𝐴) × ({1o} × 𝐵)) ≈ (𝐴 × 𝐵)) |
22 | domentr 9006 | . 2 ⊢ (((𝐴 ⊔ 𝐵) ≼ (({∅} × 𝐴) × ({1o} × 𝐵)) ∧ (({∅} × 𝐴) × ({1o} × 𝐵)) ≈ (𝐴 × 𝐵)) → (𝐴 ⊔ 𝐵) ≼ (𝐴 × 𝐵)) | |
23 | 19, 21, 22 | syl2anc 583 | 1 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 Vcvv 3466 ∪ cun 3939 ∅c0 4315 {csn 4621 class class class wbr 5139 × cxp 5665 Oncon0 6355 1oc1o 8455 ≈ cen 8933 ≼ cdom 8934 ≺ csdm 8935 ⊔ cdju 9890 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-1st 7969 df-2nd 7970 df-1o 8462 df-2o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-dju 9893 |
This theorem is referenced by: canthp1lem1 10644 |
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