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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 4632 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snex 5433 | . . . 4 ⊢ {𝐴} ∈ V | |
3 | snex 5433 | . . . 4 ⊢ {𝐵} ∈ V | |
4 | undjudom 10191 | . . . 4 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵})) | |
5 | 2, 3, 4 | mp2an 691 | . . 3 ⊢ ({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵}) |
6 | sn1dom 42956 | . . . . . 6 ⊢ {𝐴} ≼ 1o | |
7 | djudom1 10206 | . . . . . 6 ⊢ (({𝐴} ≼ 1o ∧ {𝐵} ∈ V) → ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵})) | |
8 | 6, 3, 7 | mp2an 691 | . . . . 5 ⊢ ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵}) |
9 | sn1dom 42956 | . . . . . 6 ⊢ {𝐵} ≼ 1o | |
10 | 1on 8499 | . . . . . 6 ⊢ 1o ∈ On | |
11 | djudom2 10207 | . . . . . 6 ⊢ (({𝐵} ≼ 1o ∧ 1o ∈ On) → (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o)) | |
12 | 9, 10, 11 | mp2an 691 | . . . . 5 ⊢ (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o) |
13 | domtr 9028 | . . . . 5 ⊢ ((({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵}) ∧ (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o)) → ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o)) | |
14 | 8, 12, 13 | mp2an 691 | . . . 4 ⊢ ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o) |
15 | dju1p1e2 10197 | . . . 4 ⊢ (1o ⊔ 1o) ≈ 2o | |
16 | domentr 9034 | . . . 4 ⊢ ((({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o) ∧ (1o ⊔ 1o) ≈ 2o) → ({𝐴} ⊔ {𝐵}) ≼ 2o) | |
17 | 14, 15, 16 | mp2an 691 | . . 3 ⊢ ({𝐴} ⊔ {𝐵}) ≼ 2o |
18 | domtr 9028 | . . 3 ⊢ ((({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵}) ∧ ({𝐴} ⊔ {𝐵}) ≼ 2o) → ({𝐴} ∪ {𝐵}) ≼ 2o) | |
19 | 5, 17, 18 | mp2an 691 | . 2 ⊢ ({𝐴} ∪ {𝐵}) ≼ 2o |
20 | 1, 19 | eqbrtri 5169 | 1 ⊢ {𝐴, 𝐵} ≼ 2o |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 Vcvv 3471 ∪ cun 3945 {csn 4629 {cpr 4631 class class class wbr 5148 Oncon0 6369 1oc1o 8480 2oc2o 8481 ≈ cen 8961 ≼ cdom 8962 ⊔ cdju 9922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-1st 7993 df-2nd 7994 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-dju 9925 |
This theorem is referenced by: tr3dom 42958 |
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