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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 4631 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
| 2 | prex 5437 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
| 3 | snex 5436 | . . . 4 ⊢ {𝐶} ∈ V | |
| 4 | undjudom 10208 | . . . 4 ⊢ (({𝐴, 𝐵} ∈ V ∧ {𝐶} ∈ V) → ({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶})) | |
| 5 | 2, 3, 4 | mp2an 692 | . . 3 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶}) |
| 6 | pr2dom 43540 | . . . . . 6 ⊢ {𝐴, 𝐵} ≼ 2o | |
| 7 | djudom1 10223 | . . . . . 6 ⊢ (({𝐴, 𝐵} ≼ 2o ∧ {𝐶} ∈ V) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶})) | |
| 8 | 6, 3, 7 | mp2an 692 | . . . . 5 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶}) |
| 9 | sn1dom 43539 | . . . . . 6 ⊢ {𝐶} ≼ 1o | |
| 10 | 2on 8520 | . . . . . 6 ⊢ 2o ∈ On | |
| 11 | djudom2 10224 | . . . . . 6 ⊢ (({𝐶} ≼ 1o ∧ 2o ∈ On) → (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
| 12 | 9, 10, 11 | mp2an 692 | . . . . 5 ⊢ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
| 13 | domtr 9047 | . . . . 5 ⊢ ((({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶}) ∧ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
| 14 | 8, 12, 13 | mp2an 692 | . . . 4 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
| 15 | 1on 8518 | . . . . . . 7 ⊢ 1o ∈ On | |
| 16 | onadju 10234 | . . . . . . 7 ⊢ ((2o ∈ On ∧ 1o ∈ On) → (2o +o 1o) ≈ (2o ⊔ 1o)) | |
| 17 | 10, 15, 16 | mp2an 692 | . . . . . 6 ⊢ (2o +o 1o) ≈ (2o ⊔ 1o) |
| 18 | 17 | ensymi 9044 | . . . . 5 ⊢ (2o ⊔ 1o) ≈ (2o +o 1o) |
| 19 | oa1suc 8569 | . . . . . . 7 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o) | |
| 20 | 10, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o +o 1o) = suc 2o |
| 21 | df-3o 8508 | . . . . . 6 ⊢ 3o = suc 2o | |
| 22 | 20, 21 | eqtr4i 2768 | . . . . 5 ⊢ (2o +o 1o) = 3o |
| 23 | 18, 22 | breqtri 5168 | . . . 4 ⊢ (2o ⊔ 1o) ≈ 3o |
| 24 | domentr 9053 | . . . 4 ⊢ ((({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o) ∧ (2o ⊔ 1o) ≈ 3o) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o) | |
| 25 | 14, 23, 24 | mp2an 692 | . . 3 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o |
| 26 | domtr 9047 | . . 3 ⊢ ((({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶}) ∧ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o) → ({𝐴, 𝐵} ∪ {𝐶}) ≼ 3o) | |
| 27 | 5, 25, 26 | mp2an 692 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ≼ 3o |
| 28 | 1, 27 | eqbrtri 5164 | 1 ⊢ {𝐴, 𝐵, 𝐶} ≼ 3o |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 {cpr 4628 {ctp 4630 class class class wbr 5143 Oncon0 6384 suc csuc 6386 (class class class)co 7431 1oc1o 8499 2oc2o 8500 3oc3o 8501 +o coa 8503 ≈ cen 8982 ≼ cdom 8983 ⊔ cdju 9938 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-3o 8508 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-dju 9941 |
| This theorem is referenced by: (None) |
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