Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 4532 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prex 5310 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
3 | snex 5309 | . . . 4 ⊢ {𝐶} ∈ V | |
4 | undjudom 9746 | . . . 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 40760 | . . . . . 6 ⊢ {𝐴, 𝐵} ≼ 2o | |
7 | djudom1 9761 | . . . . . 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 40759 | . . . . . 6 ⊢ {𝐶} ≼ 1o | |
10 | 2on 8188 | . . . . . 6 ⊢ 2o ∈ On | |
11 | djudom2 9762 | . . . . . 6 ⊢ (({𝐶} ≼ 1o ∧ 2o ∈ On) → (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
12 | 9, 10, 11 | mp2an 692 | . . . . 5 ⊢ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
13 | domtr 8659 | . . . . 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 8187 | . . . . . . 7 ⊢ 1o ∈ On | |
16 | onadju 9772 | . . . . . . 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 8656 | . . . . 5 ⊢ (2o ⊔ 1o) ≈ (2o +o 1o) |
19 | oa1suc 8236 | . . . . . . 7 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o) | |
20 | 10, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o +o 1o) = suc 2o |
21 | df-3o 8182 | . . . . . 6 ⊢ 3o = suc 2o | |
22 | 20, 21 | eqtr4i 2762 | . . . . 5 ⊢ (2o +o 1o) = 3o |
23 | 18, 22 | breqtri 5064 | . . . 4 ⊢ (2o ⊔ 1o) ≈ 3o |
24 | domentr 8665 | . . . 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 8659 | . . 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 5060 | 1 ⊢ {𝐴, 𝐵, 𝐶} ≼ 3o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∪ cun 3851 {csn 4527 {cpr 4529 {ctp 4531 class class class wbr 5039 Oncon0 6191 suc csuc 6193 (class class class)co 7191 1oc1o 8173 2oc2o 8174 3oc3o 8175 +o coa 8177 ≈ cen 8601 ≼ cdom 8602 ⊔ cdju 9479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-3o 8182 df-oadd 8184 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-dju 9482 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |