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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 4563 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snex 5325 | . . . 4 ⊢ {𝐴} ∈ V | |
3 | snex 5325 | . . . 4 ⊢ {𝐵} ∈ V | |
4 | undjudom 9586 | . . . 4 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵})) | |
5 | 2, 3, 4 | mp2an 690 | . . 3 ⊢ ({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵}) |
6 | sn1dom 39967 | . . . . . 6 ⊢ {𝐴} ≼ 1o | |
7 | djudom1 9601 | . . . . . 6 ⊢ (({𝐴} ≼ 1o ∧ {𝐵} ∈ V) → ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵})) | |
8 | 6, 3, 7 | mp2an 690 | . . . . 5 ⊢ ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵}) |
9 | sn1dom 39967 | . . . . . 6 ⊢ {𝐵} ≼ 1o | |
10 | 1on 8102 | . . . . . 6 ⊢ 1o ∈ On | |
11 | djudom2 9602 | . . . . . 6 ⊢ (({𝐵} ≼ 1o ∧ 1o ∈ On) → (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o)) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o) |
13 | domtr 8555 | . . . . 5 ⊢ ((({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ {𝐵}) ∧ (1o ⊔ {𝐵}) ≼ (1o ⊔ 1o)) → ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o)) | |
14 | 8, 12, 13 | mp2an 690 | . . . 4 ⊢ ({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o) |
15 | dju1p1e2 9592 | . . . 4 ⊢ (1o ⊔ 1o) ≈ 2o | |
16 | domentr 8561 | . . . 4 ⊢ ((({𝐴} ⊔ {𝐵}) ≼ (1o ⊔ 1o) ∧ (1o ⊔ 1o) ≈ 2o) → ({𝐴} ⊔ {𝐵}) ≼ 2o) | |
17 | 14, 15, 16 | mp2an 690 | . . 3 ⊢ ({𝐴} ⊔ {𝐵}) ≼ 2o |
18 | domtr 8555 | . . 3 ⊢ ((({𝐴} ∪ {𝐵}) ≼ ({𝐴} ⊔ {𝐵}) ∧ ({𝐴} ⊔ {𝐵}) ≼ 2o) → ({𝐴} ∪ {𝐵}) ≼ 2o) | |
19 | 5, 17, 18 | mp2an 690 | . 2 ⊢ ({𝐴} ∪ {𝐵}) ≼ 2o |
20 | 1, 19 | eqbrtri 5080 | 1 ⊢ {𝐴, 𝐵} ≼ 2o |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 Vcvv 3491 ∪ cun 3927 {csn 4560 {cpr 4562 class class class wbr 5059 Oncon0 6184 1oc1o 8088 2oc2o 8089 ≈ cen 8499 ≼ cdom 8500 ⊔ cdju 9320 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7574 df-1st 7682 df-2nd 7683 df-1o 8095 df-2o 8096 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-dju 9323 |
This theorem is referenced by: tr3dom 39969 |
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