![]() |
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 4592 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prex 5390 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
3 | snex 5389 | . . . 4 ⊢ {𝐶} ∈ V | |
4 | undjudom 10104 | . . . 4 ⊢ (({𝐴, 𝐵} ∈ V ∧ {𝐶} ∈ V) → ({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶})) | |
5 | 2, 3, 4 | mp2an 691 | . . 3 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶}) |
6 | pr2dom 41806 | . . . . . 6 ⊢ {𝐴, 𝐵} ≼ 2o | |
7 | djudom1 10119 | . . . . . 6 ⊢ (({𝐴, 𝐵} ≼ 2o ∧ {𝐶} ∈ V) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶})) | |
8 | 6, 3, 7 | mp2an 691 | . . . . 5 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶}) |
9 | sn1dom 41805 | . . . . . 6 ⊢ {𝐶} ≼ 1o | |
10 | 2on 8427 | . . . . . 6 ⊢ 2o ∈ On | |
11 | djudom2 10120 | . . . . . 6 ⊢ (({𝐶} ≼ 1o ∧ 2o ∈ On) → (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
12 | 9, 10, 11 | mp2an 691 | . . . . 5 ⊢ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
13 | domtr 8948 | . . . . 5 ⊢ ((({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ {𝐶}) ∧ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
14 | 8, 12, 13 | mp2an 691 | . . . 4 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
15 | 1on 8425 | . . . . . . 7 ⊢ 1o ∈ On | |
16 | onadju 10130 | . . . . . . 7 ⊢ ((2o ∈ On ∧ 1o ∈ On) → (2o +o 1o) ≈ (2o ⊔ 1o)) | |
17 | 10, 15, 16 | mp2an 691 | . . . . . 6 ⊢ (2o +o 1o) ≈ (2o ⊔ 1o) |
18 | 17 | ensymi 8945 | . . . . 5 ⊢ (2o ⊔ 1o) ≈ (2o +o 1o) |
19 | oa1suc 8478 | . . . . . . 7 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o) | |
20 | 10, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o +o 1o) = suc 2o |
21 | df-3o 8415 | . . . . . 6 ⊢ 3o = suc 2o | |
22 | 20, 21 | eqtr4i 2768 | . . . . 5 ⊢ (2o +o 1o) = 3o |
23 | 18, 22 | breqtri 5131 | . . . 4 ⊢ (2o ⊔ 1o) ≈ 3o |
24 | domentr 8954 | . . . 4 ⊢ ((({𝐴, 𝐵} ⊔ {𝐶}) ≼ (2o ⊔ 1o) ∧ (2o ⊔ 1o) ≈ 3o) → ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o) | |
25 | 14, 23, 24 | mp2an 691 | . . 3 ⊢ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o |
26 | domtr 8948 | . . 3 ⊢ ((({𝐴, 𝐵} ∪ {𝐶}) ≼ ({𝐴, 𝐵} ⊔ {𝐶}) ∧ ({𝐴, 𝐵} ⊔ {𝐶}) ≼ 3o) → ({𝐴, 𝐵} ∪ {𝐶}) ≼ 3o) | |
27 | 5, 25, 26 | mp2an 691 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) ≼ 3o |
28 | 1, 27 | eqbrtri 5127 | 1 ⊢ {𝐴, 𝐵, 𝐶} ≼ 3o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3446 ∪ cun 3909 {csn 4587 {cpr 4589 {ctp 4591 class class class wbr 5106 Oncon0 6318 suc csuc 6320 (class class class)co 7358 1oc1o 8406 2oc2o 8407 3oc3o 8408 +o coa 8410 ≈ cen 8881 ≼ cdom 8882 ⊔ cdju 9835 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-3o 8415 df-oadd 8417 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-dju 9838 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |