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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 9895 | . . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 0ex 5307 | . . . . . . 7 ⊢ ∅ ∈ V | |
3 | relsdom 8945 | . . . . . . . 8 ⊢ Rel ≺ | |
4 | 3 | brrelex2i 5733 | . . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V) |
5 | xpsnen2g 9064 | . . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
6 | 2, 4, 5 | sylancr 587 | . . . . . 6 ⊢ (1o ≺ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
7 | sdomen2 9121 | . . . . . 6 ⊢ (({∅} × 𝐴) ≈ 𝐴 → (1o ≺ ({∅} × 𝐴) ↔ 1o ≺ 𝐴)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (1o ≺ 𝐴 → (1o ≺ ({∅} × 𝐴) ↔ 1o ≺ 𝐴)) |
9 | 8 | ibir 267 | . . . 4 ⊢ (1o ≺ 𝐴 → 1o ≺ ({∅} × 𝐴)) |
10 | 1on 8477 | . . . . . . 7 ⊢ 1o ∈ On | |
11 | 3 | brrelex2i 5733 | . . . . . . 7 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V) |
12 | xpsnen2g 9064 | . . . . . . 7 ⊢ ((1o ∈ On ∧ 𝐵 ∈ V) → ({1o} × 𝐵) ≈ 𝐵) | |
13 | 10, 11, 12 | sylancr 587 | . . . . . 6 ⊢ (1o ≺ 𝐵 → ({1o} × 𝐵) ≈ 𝐵) |
14 | sdomen2 9121 | . . . . . 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 267 | . . . 4 ⊢ (1o ≺ 𝐵 → 1o ≺ ({1o} × 𝐵)) |
17 | unxpdom 9252 | . . . 4 ⊢ ((1o ≺ ({∅} × 𝐴) ∧ 1o ≺ ({1o} × 𝐵)) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) | |
18 | 9, 16, 17 | syl2an 596 | . . 3 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) |
19 | 1, 18 | eqbrtrid 5183 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) |
20 | xpen 9139 | . . 3 ⊢ ((({∅} × 𝐴) ≈ 𝐴 ∧ ({1o} × 𝐵) ≈ 𝐵) → (({∅} × 𝐴) × ({1o} × 𝐵)) ≈ (𝐴 × 𝐵)) | |
21 | 6, 13, 20 | syl2an 596 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (({∅} × 𝐴) × ({1o} × 𝐵)) ≈ (𝐴 × 𝐵)) |
22 | domentr 9008 | . 2 ⊢ (((𝐴 ⊔ 𝐵) ≼ (({∅} × 𝐴) × ({1o} × 𝐵)) ∧ (({∅} × 𝐴) × ({1o} × 𝐵)) ≈ (𝐴 × 𝐵)) → (𝐴 ⊔ 𝐵) ≼ (𝐴 × 𝐵)) | |
23 | 19, 21, 22 | syl2anc 584 | 1 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3474 ∪ cun 3946 ∅c0 4322 {csn 4628 class class class wbr 5148 × cxp 5674 Oncon0 6364 1oc1o 8458 ≈ cen 8935 ≼ cdom 8936 ≺ csdm 8937 ⊔ cdju 9892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1st 7974 df-2nd 7975 df-1o 8465 df-2o 8466 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-dju 9895 |
This theorem is referenced by: canthp1lem1 10646 |
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