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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 4636 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prex 5443 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
3 | snex 5442 | . . . 4 ⊢ {𝐶} ∈ V | |
4 | undjudom 10206 | . . . 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 43517 | . . . . . 6 ⊢ {𝐴, 𝐵} ≼ 2o | |
7 | djudom1 10221 | . . . . . 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 43516 | . . . . . 6 ⊢ {𝐶} ≼ 1o | |
10 | 2on 8519 | . . . . . 6 ⊢ 2o ∈ On | |
11 | djudom2 10222 | . . . . . 6 ⊢ (({𝐶} ≼ 1o ∧ 2o ∈ On) → (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o)) | |
12 | 9, 10, 11 | mp2an 692 | . . . . 5 ⊢ (2o ⊔ {𝐶}) ≼ (2o ⊔ 1o) |
13 | domtr 9046 | . . . . 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 8517 | . . . . . . 7 ⊢ 1o ∈ On | |
16 | onadju 10232 | . . . . . . 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 9043 | . . . . 5 ⊢ (2o ⊔ 1o) ≈ (2o +o 1o) |
19 | oa1suc 8568 | . . . . . . 7 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o) | |
20 | 10, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o +o 1o) = suc 2o |
21 | df-3o 8507 | . . . . . 6 ⊢ 3o = suc 2o | |
22 | 20, 21 | eqtr4i 2766 | . . . . 5 ⊢ (2o +o 1o) = 3o |
23 | 18, 22 | breqtri 5173 | . . . 4 ⊢ (2o ⊔ 1o) ≈ 3o |
24 | domentr 9052 | . . . 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 9046 | . . 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 5169 | 1 ⊢ {𝐴, 𝐵, 𝐶} ≼ 3o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 {csn 4631 {cpr 4633 {ctp 4635 class class class wbr 5148 Oncon0 6386 suc csuc 6388 (class class class)co 7431 1oc1o 8498 2oc2o 8499 3oc3o 8500 +o coa 8502 ≈ cen 8981 ≼ cdom 8982 ⊔ cdju 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-3o 8507 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-dju 9939 |
This theorem is referenced by: (None) |
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