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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 5211 | . . . . 5 ⊢ ∅ ∈ V | |
2 | xpsnen2g 8610 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ≈ 𝐴) | |
3 | 1, 2 | mpan 688 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} × 𝐴) ≈ 𝐴) |
4 | ensym 8558 | . . . 4 ⊢ (({∅} × 𝐴) ≈ 𝐴 → 𝐴 ≈ ({∅} × 𝐴)) | |
5 | endom 8536 | . . . 4 ⊢ (𝐴 ≈ ({∅} × 𝐴) → 𝐴 ≼ ({∅} × 𝐴)) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ ({∅} × 𝐴)) |
7 | 1on 8109 | . . . . 5 ⊢ 1o ∈ On | |
8 | xpsnen2g 8610 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ≈ 𝐵) | |
9 | 7, 8 | mpan 688 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ({1o} × 𝐵) ≈ 𝐵) |
10 | ensym 8558 | . . . 4 ⊢ (({1o} × 𝐵) ≈ 𝐵 → 𝐵 ≈ ({1o} × 𝐵)) | |
11 | endom 8536 | . . . 4 ⊢ (𝐵 ≈ ({1o} × 𝐵) → 𝐵 ≼ ({1o} × 𝐵)) | |
12 | 9, 10, 11 | 3syl 18 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ≼ ({1o} × 𝐵)) |
13 | xp01disjl 8121 | . . . 4 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ | |
14 | undom 8605 | . . . 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 9330 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
18 | 16, 17 | breqtrrdi 5108 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 {csn 4567 class class class wbr 5066 × cxp 5553 Oncon0 6191 1oc1o 8095 ≈ cen 8506 ≼ cdom 8507 ⊔ cdju 9327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-1st 7689 df-2nd 7690 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-dju 9330 |
This theorem is referenced by: djudoml 9610 unnum 9622 ficardun2 9625 pwsdompw 9626 unctb 9627 infunabs 9629 infdju 9630 infdif 9631 pr2dom 39942 tr3dom 39943 |
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