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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 5305 | . . . . 5 ⊢ ∅ ∈ V | |
2 | xpsnen2g 9060 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ≈ 𝐴) | |
3 | 1, 2 | mpan 689 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} × 𝐴) ≈ 𝐴) |
4 | ensym 8994 | . . . 4 ⊢ (({∅} × 𝐴) ≈ 𝐴 → 𝐴 ≈ ({∅} × 𝐴)) | |
5 | endom 8970 | . . . 4 ⊢ (𝐴 ≈ ({∅} × 𝐴) → 𝐴 ≼ ({∅} × 𝐴)) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ ({∅} × 𝐴)) |
7 | 1on 8472 | . . . . 5 ⊢ 1o ∈ On | |
8 | xpsnen2g 9060 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ≈ 𝐵) | |
9 | 7, 8 | mpan 689 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ({1o} × 𝐵) ≈ 𝐵) |
10 | ensym 8994 | . . . 4 ⊢ (({1o} × 𝐵) ≈ 𝐵 → 𝐵 ≈ ({1o} × 𝐵)) | |
11 | endom 8970 | . . . 4 ⊢ (𝐵 ≈ ({1o} × 𝐵) → 𝐵 ≼ ({1o} × 𝐵)) | |
12 | 9, 10, 11 | 3syl 18 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ≼ ({1o} × 𝐵)) |
13 | xp01disjl 8486 | . . . 4 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ | |
14 | undom 9054 | . . . 4 ⊢ (((𝐴 ≼ ({∅} × 𝐴) ∧ 𝐵 ≼ ({1o} × 𝐵)) ∧ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅) → (𝐴 ∪ 𝐵) ≼ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
15 | 13, 14 | mpan2 690 | . . 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 9891 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
18 | 16, 17 | breqtrrdi 5188 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3944 ∩ cin 3945 ∅c0 4320 {csn 4626 class class class wbr 5146 × cxp 5672 Oncon0 6360 1oc1o 8453 ≈ cen 8931 ≼ cdom 8932 ⊔ cdju 9888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-ord 6363 df-on 6364 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-1st 7969 df-2nd 7970 df-1o 8460 df-er 8698 df-en 8935 df-dom 8936 df-dju 9891 |
This theorem is referenced by: djudoml 10174 unnum 10186 ficardun2 10192 ficardun2OLD 10193 pwsdompw 10194 unctb 10195 infunabs 10197 infdju 10198 infdif 10199 pr2dom 42210 tr3dom 42211 |
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