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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 9396 | . . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 0ex 5172 | . . . . . . 7 ⊢ ∅ ∈ V | |
3 | relsdom 8555 | . . . . . . . 8 ⊢ Rel ≺ | |
4 | 3 | brrelex2i 5574 | . . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V) |
5 | xpsnen2g 8652 | . . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
6 | 2, 4, 5 | sylancr 590 | . . . . . 6 ⊢ (1o ≺ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
7 | sdomen2 8705 | . . . . . 6 ⊢ (({∅} × 𝐴) ≈ 𝐴 → (1o ≺ ({∅} × 𝐴) ↔ 1o ≺ 𝐴)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (1o ≺ 𝐴 → (1o ≺ ({∅} × 𝐴) ↔ 1o ≺ 𝐴)) |
9 | 8 | ibir 271 | . . . 4 ⊢ (1o ≺ 𝐴 → 1o ≺ ({∅} × 𝐴)) |
10 | 1on 8131 | . . . . . . 7 ⊢ 1o ∈ On | |
11 | 3 | brrelex2i 5574 | . . . . . . 7 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V) |
12 | xpsnen2g 8652 | . . . . . . 7 ⊢ ((1o ∈ On ∧ 𝐵 ∈ V) → ({1o} × 𝐵) ≈ 𝐵) | |
13 | 10, 11, 12 | sylancr 590 | . . . . . 6 ⊢ (1o ≺ 𝐵 → ({1o} × 𝐵) ≈ 𝐵) |
14 | sdomen2 8705 | . . . . . 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 271 | . . . 4 ⊢ (1o ≺ 𝐵 → 1o ≺ ({1o} × 𝐵)) |
17 | unxpdom 8797 | . . . 4 ⊢ ((1o ≺ ({∅} × 𝐴) ∧ 1o ≺ ({1o} × 𝐵)) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) | |
18 | 9, 16, 17 | syl2an 599 | . . 3 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) |
19 | 1, 18 | eqbrtrid 5062 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) |
20 | xpen 8723 | . . 3 ⊢ ((({∅} × 𝐴) ≈ 𝐴 ∧ ({1o} × 𝐵) ≈ 𝐵) → (({∅} × 𝐴) × ({1o} × 𝐵)) ≈ (𝐴 × 𝐵)) | |
21 | 6, 13, 20 | syl2an 599 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (({∅} × 𝐴) × ({1o} × 𝐵)) ≈ (𝐴 × 𝐵)) |
22 | domentr 8607 | . 2 ⊢ (((𝐴 ⊔ 𝐵) ≼ (({∅} × 𝐴) × ({1o} × 𝐵)) ∧ (({∅} × 𝐴) × ({1o} × 𝐵)) ≈ (𝐴 × 𝐵)) → (𝐴 ⊔ 𝐵) ≼ (𝐴 × 𝐵)) | |
23 | 19, 21, 22 | syl2anc 587 | 1 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2113 Vcvv 3397 ∪ cun 3839 ∅c0 4209 {csn 4513 class class class wbr 5027 × cxp 5517 Oncon0 6166 1oc1o 8117 ≈ cen 8545 ≼ cdom 8546 ≺ csdm 8547 ⊔ cdju 9393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-om 7594 df-1st 7707 df-2nd 7708 df-1o 8124 df-2o 8125 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-dju 9396 |
This theorem is referenced by: canthp1lem1 10145 |
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