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Mirrors > Home > MPE Home > Th. List > rankelun | Structured version Visualization version GIF version |
Description: Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankelun.1 | ⊢ 𝐴 ∈ V |
rankelun.2 | ⊢ 𝐵 ∈ V |
rankelun.3 | ⊢ 𝐶 ∈ V |
rankelun.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
rankelun | ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankon 9205 | . . . . 5 ⊢ (rank‘𝐶) ∈ On | |
2 | rankon 9205 | . . . . 5 ⊢ (rank‘𝐷) ∈ On | |
3 | 1, 2 | onun2i 6287 | . . . 4 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On |
4 | 3 | onordi 6276 | . . 3 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷)) |
5 | elun1 4135 | . . 3 ⊢ ((rank‘𝐴) ∈ (rank‘𝐶) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) | |
6 | elun2 4136 | . . 3 ⊢ ((rank‘𝐵) ∈ (rank‘𝐷) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) | |
7 | ordunel 7523 | . . 3 ⊢ ((Ord ((rank‘𝐶) ∪ (rank‘𝐷)) ∧ (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)) ∧ (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) | |
8 | 4, 5, 6, 7 | mp3an3an 1463 | . 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) |
9 | rankelun.1 | . . 3 ⊢ 𝐴 ∈ V | |
10 | rankelun.2 | . . 3 ⊢ 𝐵 ∈ V | |
11 | 9, 10 | rankun 9266 | . 2 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)) |
12 | rankelun.3 | . . 3 ⊢ 𝐶 ∈ V | |
13 | rankelun.4 | . . 3 ⊢ 𝐷 ∈ V | |
14 | 12, 13 | rankun 9266 | . 2 ⊢ (rank‘(𝐶 ∪ 𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷)) |
15 | 8, 11, 14 | 3eltr4g 2928 | 1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 Vcvv 3481 ∪ cun 3917 Ord word 6171 ‘cfv 6336 rankcrnk 9173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-reg 9037 ax-inf2 9085 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-om 7562 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-r1 9174 df-rank 9175 |
This theorem is referenced by: rankelpr 9283 rankxplim 9289 |
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