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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 9941 | . . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
| 2 | 0ex 5307 | . . . . . . 7 ⊢ ∅ ∈ V | |
| 3 | relsdom 8992 | . . . . . . . 8 ⊢ Rel ≺ | |
| 4 | 3 | brrelex2i 5742 | . . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V) |
| 5 | xpsnen2g 9105 | . . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
| 6 | 2, 4, 5 | sylancr 587 | . . . . . 6 ⊢ (1o ≺ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
| 7 | sdomen2 9162 | . . . . . 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 8518 | . . . . . . 7 ⊢ 1o ∈ On | |
| 11 | 3 | brrelex2i 5742 | . . . . . . 7 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V) |
| 12 | xpsnen2g 9105 | . . . . . . 7 ⊢ ((1o ∈ On ∧ 𝐵 ∈ V) → ({1o} × 𝐵) ≈ 𝐵) | |
| 13 | 10, 11, 12 | sylancr 587 | . . . . . 6 ⊢ (1o ≺ 𝐵 → ({1o} × 𝐵) ≈ 𝐵) |
| 14 | sdomen2 9162 | . . . . . 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 9289 | . . . 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 5178 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) |
| 20 | xpen 9180 | . . 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 9053 | . 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 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∅c0 4333 {csn 4626 class class class wbr 5143 × cxp 5683 Oncon0 6384 1oc1o 8499 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 ⊔ cdju 9938 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-dju 9941 |
| This theorem is referenced by: canthp1lem1 10692 |
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