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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 9924 | . . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 0ex 5307 | . . . . . . 7 ⊢ ∅ ∈ V | |
3 | relsdom 8970 | . . . . . . . 8 ⊢ Rel ≺ | |
4 | 3 | brrelex2i 5735 | . . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V) |
5 | xpsnen2g 9089 | . . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
6 | 2, 4, 5 | sylancr 586 | . . . . . 6 ⊢ (1o ≺ 𝐴 → ({∅} × 𝐴) ≈ 𝐴) |
7 | sdomen2 9146 | . . . . . 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 8498 | . . . . . . 7 ⊢ 1o ∈ On | |
11 | 3 | brrelex2i 5735 | . . . . . . 7 ⊢ (1o ≺ 𝐵 → 𝐵 ∈ V) |
12 | xpsnen2g 9089 | . . . . . . 7 ⊢ ((1o ∈ On ∧ 𝐵 ∈ V) → ({1o} × 𝐵) ≈ 𝐵) | |
13 | 10, 11, 12 | sylancr 586 | . . . . . 6 ⊢ (1o ≺ 𝐵 → ({1o} × 𝐵) ≈ 𝐵) |
14 | sdomen2 9146 | . . . . . 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 9277 | . . . 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 5183 | . 2 ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (({∅} × 𝐴) × ({1o} × 𝐵))) |
20 | xpen 9164 | . . 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 9033 | . 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 2099 Vcvv 3471 ∪ cun 3945 ∅c0 4323 {csn 4629 class class class wbr 5148 × cxp 5676 Oncon0 6369 1oc1o 8479 ≈ cen 8960 ≼ cdom 8961 ≺ csdm 8962 ⊔ cdju 9921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-1st 7993 df-2nd 7994 df-1o 8486 df-2o 8487 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-dju 9924 |
This theorem is referenced by: canthp1lem1 10675 |
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