MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankelun Structured version   Visualization version   GIF version

Theorem rankelun 8682
Description: Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)
Hypotheses
Ref Expression
rankelun.1 𝐴 ∈ V
rankelun.2 𝐵 ∈ V
rankelun.3 𝐶 ∈ V
rankelun.4 𝐷 ∈ V
Assertion
Ref Expression
rankelun (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴𝐵)) ∈ (rank‘(𝐶𝐷)))

Proof of Theorem rankelun
StepHypRef Expression
1 rankon 8605 . . . . . 6 (rank‘𝐶) ∈ On
2 rankon 8605 . . . . . 6 (rank‘𝐷) ∈ On
31, 2onun2i 5804 . . . . 5 ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
43onordi 5793 . . . 4 Ord ((rank‘𝐶) ∪ (rank‘𝐷))
54a1i 11 . . 3 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → Ord ((rank‘𝐶) ∪ (rank‘𝐷)))
6 elun1 3760 . . . 4 ((rank‘𝐴) ∈ (rank‘𝐶) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
76adantr 481 . . 3 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
8 elun2 3761 . . . 4 ((rank‘𝐵) ∈ (rank‘𝐷) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
98adantl 482 . . 3 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
10 ordunel 6977 . . 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)))
115, 7, 9, 10syl3anc 1323 . 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
1412, 13rankun 8666 . 2 (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
15 rankelun.3 . . 3 𝐶 ∈ V
16 rankelun.4 . . 3 𝐷 ∈ V
1715, 16rankun 8666 . 2 (rank‘(𝐶𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
1811, 14, 173eltr4g 2715 1 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴𝐵)) ∈ (rank‘(𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Vcvv 3186  cun 3554  Ord word 5683  cfv 5849  rankcrnk 8573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-reg 8444  ax-inf2 8485
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-om 7016  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-r1 8574  df-rank 8575
This theorem is referenced by:  rankelpr  8683  rankxplim  8689
  Copyright terms: Public domain W3C validator