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| Mirrors > Home > MPE Home > Th. List > undjudom | Structured version Visualization version GIF version | ||
| Description: Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.) (Revised by Jim Kingdon, 15-Aug-2023.) |
| Ref | Expression |
|---|---|
| undjudom | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 2 | xpsnen2g 9105 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ≈ 𝐴) | |
| 3 | 1, 2 | mpan 690 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} × 𝐴) ≈ 𝐴) |
| 4 | ensym 9043 | . . . 4 ⊢ (({∅} × 𝐴) ≈ 𝐴 → 𝐴 ≈ ({∅} × 𝐴)) | |
| 5 | endom 9019 | . . . 4 ⊢ (𝐴 ≈ ({∅} × 𝐴) → 𝐴 ≼ ({∅} × 𝐴)) | |
| 6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ ({∅} × 𝐴)) |
| 7 | 1on 8518 | . . . . 5 ⊢ 1o ∈ On | |
| 8 | xpsnen2g 9105 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ≈ 𝐵) | |
| 9 | 7, 8 | mpan 690 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ({1o} × 𝐵) ≈ 𝐵) |
| 10 | ensym 9043 | . . . 4 ⊢ (({1o} × 𝐵) ≈ 𝐵 → 𝐵 ≈ ({1o} × 𝐵)) | |
| 11 | endom 9019 | . . . 4 ⊢ (𝐵 ≈ ({1o} × 𝐵) → 𝐵 ≼ ({1o} × 𝐵)) | |
| 12 | 9, 10, 11 | 3syl 18 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ≼ ({1o} × 𝐵)) |
| 13 | xp01disjl 8530 | . . . 4 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ | |
| 14 | undom 9099 | . . . 4 ⊢ (((𝐴 ≼ ({∅} × 𝐴) ∧ 𝐵 ≼ ({1o} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅) → (𝐴 ∪ 𝐵) ≼ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
| 15 | 13, 14 | mpan2 691 | . . 3 ⊢ ((𝐴 ≼ ({∅} × 𝐴) ∧ 𝐵 ≼ ({1o} × 𝐵)) → (𝐴 ∪ 𝐵) ≼ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
| 16 | 6, 12, 15 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ≼ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
| 17 | df-dju 9941 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
| 18 | 16, 17 | breqtrrdi 5185 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 class class class wbr 5143 × cxp 5683 Oncon0 6384 1oc1o 8499 ≈ cen 8982 ≼ cdom 8983 ⊔ 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-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-er 8745 df-en 8986 df-dom 8987 df-dju 9941 |
| This theorem is referenced by: djudoml 10225 unnum 10237 ficardun2 10242 pwsdompw 10243 unctb 10244 infunabs 10246 infdju 10247 infdif 10248 pr2dom 43540 tr3dom 43541 |
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