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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 4628 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prex 5425 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
3 | snex 5424 | . . . 4 ⊢ {𝐶} ∈ V | |
4 | undjudom 10161 | . . . 4 ⊢ (({𝐴, 𝐵} ∈ V ∧ {𝐶} ∈ V) → ({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶})) | |
5 | 2, 3, 4 | mp2an 689 | . . 3 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶}) |
6 | pr2dom 42835 | . . . . . 6 ⊢ {𝐴, 𝐵} ≼ 2o | |
7 | djudom1 10176 | . . . . . 6 ⊢ (({𝐴, 𝐵} ≼ 2o ∧ {𝐶} ∈ V) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶})) | |
8 | 6, 3, 7 | mp2an 689 | . . . . 5 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶}) |
9 | sn1dom 42834 | . . . . . 6 ⊢ {𝐶} ≼ 1o | |
10 | 2on 8478 | . . . . . 6 ⊢ 2o ∈ On | |
11 | djudom2 10177 | . . . . . 6 ⊢ (({𝐶} ≼ 1o ∧ 2o ∈ On) → (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
12 | 9, 10, 11 | mp2an 689 | . . . . 5 ⊢ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
13 | domtr 9002 | . . . . 5 ⊢ ((({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶}) ∧ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
14 | 8, 12, 13 | mp2an 689 | . . . 4 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
15 | 1on 8476 | . . . . . . 7 ⊢ 1o ∈ On | |
16 | onadju 10187 | . . . . . . 7 ⊢ ((2o ∈ On ∧ 1o ∈ On) → (2o +o 1o) ≈ (2o ⊔ 1o)) | |
17 | 10, 15, 16 | mp2an 689 | . . . . . 6 ⊢ (2o +o 1o) ≈ (2o ⊔ 1o) |
18 | 17 | ensymi 8999 | . . . . 5 ⊢ (2o ⊔ 1o) ≈ (2o +o 1o) |
19 | oa1suc 8529 | . . . . . . 7 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o) | |
20 | 10, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o +o 1o) = suc 2o |
21 | df-3o 8466 | . . . . . 6 ⊢ 3o = suc 2o | |
22 | 20, 21 | eqtr4i 2757 | . . . . 5 ⊢ (2o +o 1o) = 3o |
23 | 18, 22 | breqtri 5166 | . . . 4 ⊢ (2o ⊔ 1o) ≈ 3o |
24 | domentr 9008 | . . . 4 ⊢ ((({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o) ∧ (2o ⊔ 1o) ≈ 3o) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o) | |
25 | 14, 23, 24 | mp2an 689 | . . 3 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o |
26 | domtr 9002 | . . 3 ⊢ ((({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶}) ∧ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o) → ({𝐴, 𝐵} ∪ {𝐶}) ≼ 3o) | |
27 | 5, 25, 26 | mp2an 689 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ≼ 3o |
28 | 1, 27 | eqbrtri 5162 | 1 ⊢ {𝐴, 𝐵, 𝐶} ≼ 3o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∪ cun 3941 {csn 4623 {cpr 4625 {ctp 4627 class class class wbr 5141 Oncon0 6357 suc csuc 6359 (class class class)co 7404 1oc1o 8457 2oc2o 8458 3oc3o 8459 +o coa 8461 ≈ cen 8935 ≼ cdom 8936 ⊔ cdju 9892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-3o 8466 df-oadd 8468 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-dju 9895 |
This theorem is referenced by: (None) |
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