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Theorem rankelpr 9290
Description: Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised 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
rankelpr (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))

Proof of Theorem rankelpr
StepHypRef Expression
1 rankelun.1 . . . . 5 𝐴 ∈ V
2 rankelun.2 . . . . 5 𝐵 ∈ V
3 rankelun.3 . . . . 5 𝐶 ∈ V
4 rankelun.4 . . . . 5 𝐷 ∈ V
51, 2, 3, 4rankelun 9289 . . . 4 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴𝐵)) ∈ (rank‘(𝐶𝐷)))
61, 2rankun 9273 . . . 4 (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
73, 4rankun 9273 . . . 4 (rank‘(𝐶𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
85, 6, 73eltr3g 2930 . . 3 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
9 rankon 9212 . . . . . 6 (rank‘𝐶) ∈ On
10 rankon 9212 . . . . . 6 (rank‘𝐷) ∈ On
119, 10onun2i 6284 . . . . 5 ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
1211onordi 6273 . . . 4 Ord ((rank‘𝐶) ∪ (rank‘𝐷))
13 ordsucelsuc 7522 . . . 4 (Ord ((rank‘𝐶) ∪ (rank‘𝐷)) → (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)) ↔ suc ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ suc ((rank‘𝐶) ∪ (rank‘𝐷))))
1412, 13ax-mp 5 . . 3 (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)) ↔ suc ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ suc ((rank‘𝐶) ∪ (rank‘𝐷)))
158, 14sylib 221 . 2 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ suc ((rank‘𝐶) ∪ (rank‘𝐷)))
161, 2rankpr 9274 . 2 (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))
173, 4rankpr 9274 . 2 (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷))
1815, 16, 173eltr4g 2931 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  wb 209  wa 399  wcel 2114  Vcvv 3469  cun 3906  {cpr 4541  Ord word 6168  suc csuc 6171  cfv 6334  rankcrnk 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-r1 9181  df-rank 9182
This theorem is referenced by:  rankelop  9291
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