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Theorem rankelun 8773
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 8696 . . . . . 6 (rank‘𝐶) ∈ On
2 rankon 8696 . . . . . 6 (rank‘𝐷) ∈ On
31, 2onun2i 5881 . . . . 5 ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
43onordi 5870 . . . 4 Ord ((rank‘𝐶) ∪ (rank‘𝐷))
54a1i 11 . . 3 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → Ord ((rank‘𝐶) ∪ (rank‘𝐷)))
6 elun1 3813 . . . 4 ((rank‘𝐴) ∈ (rank‘𝐶) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
76adantr 480 . . 3 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
8 elun2 3814 . . . 4 ((rank‘𝐵) ∈ (rank‘𝐷) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
98adantl 481 . . 3 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
10 ordunel 7069 . . 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 1366 . 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 8757 . 2 (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
15 rankelun.3 . . 3 𝐶 ∈ V
16 rankelun.4 . . 3 𝐷 ∈ V
1715, 16rankun 8757 . 2 (rank‘(𝐶𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
1811, 14, 173eltr4g 2747 1 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴𝐵)) ∈ (rank‘(𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  Vcvv 3231  cun 3605  Ord word 5760  cfv 5926  rankcrnk 8664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666
This theorem is referenced by:  rankelpr  8774  rankxplim  8780
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