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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pr2dom | Structured version Visualization version GIF version | ||
| Description: An unordered pair is dominated by ordinal two. (Contributed by RP, 29-Oct-2023.) |
| Ref | Expression |
|---|---|
| pr2dom | ⊢ {𝐴, 𝐵} ≼ 2o |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-pr 4629 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 2 | snex 5436 | . . . 4 ⊢ {𝐴} ∈ V | |
| 3 | snex 5436 | . . . 4 ⊢ {𝐵} ∈ V | |
| 4 | undjudom 10208 | . . . 4 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵})) | |
| 5 | 2, 3, 4 | mp2an 692 | . . 3 ⊢ ({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵}) |
| 6 | sn1dom 43539 | . . . . . 6 ⊢ {𝐴} ≼ 1o | |
| 7 | djudom1 10223 | . . . . . 6 ⊢ (({𝐴} ≼ 1o ∧ {𝐵} ∈ V) → ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵})) | |
| 8 | 6, 3, 7 | mp2an 692 | . . . . 5 ⊢ ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵}) |
| 9 | sn1dom 43539 | . . . . . 6 ⊢ {𝐵} ≼ 1o | |
| 10 | 1on 8518 | . . . . . 6 ⊢ 1o ∈ On | |
| 11 | djudom2 10224 | . . . . . 6 ⊢ (({𝐵} ≼ 1o ∧ 1o ∈ On) → (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o)) | |
| 12 | 9, 10, 11 | mp2an 692 | . . . . 5 ⊢ (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o) |
| 13 | domtr 9047 | . . . . 5 ⊢ ((({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵}) ∧ (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o)) → ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o)) | |
| 14 | 8, 12, 13 | mp2an 692 | . . . 4 ⊢ ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o) |
| 15 | dju1p1e2 10214 | . . . 4 ⊢ (1o ⊔ 1o) ≈ 2o | |
| 16 | domentr 9053 | . . . 4 ⊢ ((({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o) ∧ (1o ⊔ 1o) ≈ 2o) → ({𝐴} ⊔ {𝐵}) ≼ 2o) | |
| 17 | 14, 15, 16 | mp2an 692 | . . 3 ⊢ ({𝐴} ⊔ {𝐵}) ≼ 2o |
| 18 | domtr 9047 | . . 3 ⊢ ((({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵}) ∧ ({𝐴} ⊔ {𝐵}) ≼ 2o) → ({𝐴} ∪ {𝐵}) ≼ 2o) | |
| 19 | 5, 17, 18 | mp2an 692 | . 2 ⊢ ({𝐴} ∪ {𝐵}) ≼ 2o |
| 20 | 1, 19 | eqbrtri 5164 | 1 ⊢ {𝐴, 𝐵} ≼ 2o |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 {cpr 4628 class class class wbr 5143 Oncon0 6384 1oc1o 8499 2oc2o 8500 ≈ 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-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-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-op 4633 df-uni 4908 df-int 4947 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-ord 6387 df-on 6388 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-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-dju 9941 |
| This theorem is referenced by: tr3dom 43541 |
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