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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 8935 | . . . . . 6 ⊢ (rank‘𝐶) ∈ On | |
2 | rankon 8935 | . . . . . 6 ⊢ (rank‘𝐷) ∈ On | |
3 | 1, 2 | onun2i 6078 | . . . . 5 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On |
4 | 3 | onordi 6067 | . . . 4 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷)) |
5 | 4 | a1i 11 | . . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → Ord ((rank‘𝐶) ∪ (rank‘𝐷))) |
6 | elun1 4007 | . . . 4 ⊢ ((rank‘𝐴) ∈ (rank‘𝐶) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) | |
7 | 6 | adantr 474 | . . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) |
8 | elun2 4008 | . . . 4 ⊢ ((rank‘𝐵) ∈ (rank‘𝐷) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) | |
9 | 8 | adantl 475 | . . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) |
10 | ordunel 7288 | . . 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))) | |
11 | 5, 7, 9, 10 | syl3anc 1496 | . 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) |
12 | rankelun.1 | . . 3 ⊢ 𝐴 ∈ V | |
13 | rankelun.2 | . . 3 ⊢ 𝐵 ∈ V | |
14 | 12, 13 | rankun 8996 | . 2 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)) |
15 | rankelun.3 | . . 3 ⊢ 𝐶 ∈ V | |
16 | rankelun.4 | . . 3 ⊢ 𝐷 ∈ V | |
17 | 15, 16 | rankun 8996 | . 2 ⊢ (rank‘(𝐶 ∪ 𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷)) |
18 | 11, 14, 17 | 3eltr4g 2923 | 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 386 ∈ wcel 2166 Vcvv 3414 ∪ cun 3796 Ord word 5962 ‘cfv 6123 rankcrnk 8903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-reg 8766 ax-inf2 8815 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-r1 8904 df-rank 8905 |
This theorem is referenced by: rankelpr 9013 rankxplim 9019 |
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