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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 5208 | . . . . 5 ⊢ ∅ ∈ V | |
2 | xpsnen2g 8607 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ≈ 𝐴) | |
3 | 1, 2 | mpan 688 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} × 𝐴) ≈ 𝐴) |
4 | ensym 8555 | . . . 4 ⊢ (({∅} × 𝐴) ≈ 𝐴 → 𝐴 ≈ ({∅} × 𝐴)) | |
5 | endom 8533 | . . . 4 ⊢ (𝐴 ≈ ({∅} × 𝐴) → 𝐴 ≼ ({∅} × 𝐴)) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ ({∅} × 𝐴)) |
7 | 1on 8106 | . . . . 5 ⊢ 1o ∈ On | |
8 | xpsnen2g 8607 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ≈ 𝐵) | |
9 | 7, 8 | mpan 688 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ({1o} × 𝐵) ≈ 𝐵) |
10 | ensym 8555 | . . . 4 ⊢ (({1o} × 𝐵) ≈ 𝐵 → 𝐵 ≈ ({1o} × 𝐵)) | |
11 | endom 8533 | . . . 4 ⊢ (𝐵 ≈ ({1o} × 𝐵) → 𝐵 ≼ ({1o} × 𝐵)) | |
12 | 9, 10, 11 | 3syl 18 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ≼ ({1o} × 𝐵)) |
13 | xp01disjl 8118 | . . . 4 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ | |
14 | undom 8602 | . . . 4 ⊢ (((𝐴 ≼ ({∅} × 𝐴) ∧ 𝐵 ≼ ({1o} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅) → (𝐴 ∪ 𝐵) ≼ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
15 | 13, 14 | mpan2 689 | . . 3 ⊢ ((𝐴 ≼ ({∅} × 𝐴) ∧ 𝐵 ≼ ({1o} × 𝐵)) → (𝐴 ∪ 𝐵) ≼ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
16 | 6, 12, 15 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ≼ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
17 | df-dju 9327 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
18 | 16, 17 | breqtrrdi 5105 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3493 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 {csn 4564 class class class wbr 5063 × cxp 5550 Oncon0 6188 1oc1o 8092 ≈ cen 8503 ≼ cdom 8504 ⊔ cdju 9324 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-ord 6191 df-on 6192 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-1st 7686 df-2nd 7687 df-1o 8099 df-er 8286 df-en 8507 df-dom 8508 df-dju 9327 |
This theorem is referenced by: djudoml 9607 unnum 9619 ficardun2 9622 pwsdompw 9623 unctb 9624 infunabs 9626 infdju 9627 infdif 9628 pr2dom 39967 tr3dom 39968 |
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