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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 4590 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snex 5389 | . . . 4 ⊢ {𝐴} ∈ V | |
3 | snex 5389 | . . . 4 ⊢ {𝐵} ∈ V | |
4 | undjudom 10104 | . . . 4 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵})) | |
5 | 2, 3, 4 | mp2an 691 | . . 3 ⊢ ({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵}) |
6 | sn1dom 41805 | . . . . . 6 ⊢ {𝐴} ≼ 1o | |
7 | djudom1 10119 | . . . . . 6 ⊢ (({𝐴} ≼ 1o ∧ {𝐵} ∈ V) → ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵})) | |
8 | 6, 3, 7 | mp2an 691 | . . . . 5 ⊢ ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵}) |
9 | sn1dom 41805 | . . . . . 6 ⊢ {𝐵} ≼ 1o | |
10 | 1on 8425 | . . . . . 6 ⊢ 1o ∈ On | |
11 | djudom2 10120 | . . . . . 6 ⊢ (({𝐵} ≼ 1o ∧ 1o ∈ On) → (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o)) | |
12 | 9, 10, 11 | mp2an 691 | . . . . 5 ⊢ (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o) |
13 | domtr 8948 | . . . . 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 10110 | . . . 4 ⊢ (1o ⊔ 1o) ≈ 2o | |
16 | domentr 8954 | . . . 4 ⊢ ((({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o) ∧ (1o ⊔ 1o) ≈ 2o) → ({𝐴} ⊔ {𝐵}) ≼ 2o) | |
17 | 14, 15, 16 | mp2an 691 | . . 3 ⊢ ({𝐴} ⊔ {𝐵}) ≼ 2o |
18 | domtr 8948 | . . 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 5127 | 1 ⊢ {𝐴, 𝐵} ≼ 2o |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3446 ∪ cun 3909 {csn 4587 {cpr 4589 class class class wbr 5106 Oncon0 6318 1oc1o 8406 2oc2o 8407 ≈ 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-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-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-op 4594 df-uni 4867 df-int 4909 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-ord 6321 df-on 6322 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-1st 7922 df-2nd 7923 df-1o 8413 df-2o 8414 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-dju 9838 |
This theorem is referenced by: tr3dom 41807 |
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