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Mirrors > Home > MPE Home > Th. List > Mathboxes > tr3dom | Structured version Visualization version GIF version |
Description: An unordered triple is dominated by ordinal three. (Contributed by RP, 29-Oct-2023.) |
Ref | Expression |
---|---|
tr3dom | ⊢ {𝐴, 𝐵, 𝐶} ≼ 3o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4632 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prex 5431 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
3 | snex 5430 | . . . 4 ⊢ {𝐶} ∈ V | |
4 | undjudom 10158 | . . . 4 ⊢ (({𝐴, 𝐵} ∈ V ∧ {𝐶} ∈ V) → ({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶})) | |
5 | 2, 3, 4 | mp2an 690 | . . 3 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶}) |
6 | pr2dom 42263 | . . . . . 6 ⊢ {𝐴, 𝐵} ≼ 2o | |
7 | djudom1 10173 | . . . . . 6 ⊢ (({𝐴, 𝐵} ≼ 2o ∧ {𝐶} ∈ V) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶})) | |
8 | 6, 3, 7 | mp2an 690 | . . . . 5 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶}) |
9 | sn1dom 42262 | . . . . . 6 ⊢ {𝐶} ≼ 1o | |
10 | 2on 8476 | . . . . . 6 ⊢ 2o ∈ On | |
11 | djudom2 10174 | . . . . . 6 ⊢ (({𝐶} ≼ 1o ∧ 2o ∈ On) → (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
13 | domtr 8999 | . . . . 5 ⊢ ((({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶}) ∧ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
14 | 8, 12, 13 | mp2an 690 | . . . 4 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
15 | 1on 8474 | . . . . . . 7 ⊢ 1o ∈ On | |
16 | onadju 10184 | . . . . . . 7 ⊢ ((2o ∈ On ∧ 1o ∈ On) → (2o +o 1o) ≈ (2o ⊔ 1o)) | |
17 | 10, 15, 16 | mp2an 690 | . . . . . 6 ⊢ (2o +o 1o) ≈ (2o ⊔ 1o) |
18 | 17 | ensymi 8996 | . . . . 5 ⊢ (2o ⊔ 1o) ≈ (2o +o 1o) |
19 | oa1suc 8527 | . . . . . . 7 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o) | |
20 | 10, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o +o 1o) = suc 2o |
21 | df-3o 8464 | . . . . . 6 ⊢ 3o = suc 2o | |
22 | 20, 21 | eqtr4i 2763 | . . . . 5 ⊢ (2o +o 1o) = 3o |
23 | 18, 22 | breqtri 5172 | . . . 4 ⊢ (2o ⊔ 1o) ≈ 3o |
24 | domentr 9005 | . . . 4 ⊢ ((({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o) ∧ (2o ⊔ 1o) ≈ 3o) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o) | |
25 | 14, 23, 24 | mp2an 690 | . . 3 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o |
26 | domtr 8999 | . . 3 ⊢ ((({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶}) ∧ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o) → ({𝐴, 𝐵} ∪ {𝐶}) ≼ 3o) | |
27 | 5, 25, 26 | mp2an 690 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ≼ 3o |
28 | 1, 27 | eqbrtri 5168 | 1 ⊢ {𝐴, 𝐵, 𝐶} ≼ 3o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∪ cun 3945 {csn 4627 {cpr 4629 {ctp 4631 class class class wbr 5147 Oncon0 6361 suc csuc 6363 (class class class)co 7405 1oc1o 8455 2oc2o 8456 3oc3o 8457 +o coa 8459 ≈ cen 8932 ≼ cdom 8933 ⊔ cdju 9889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-3o 8464 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-dju 9892 |
This theorem is referenced by: (None) |
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