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Theorem rankelop 9287
Description: Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.)
Hypotheses
Ref Expression
rankelun.1 𝐴 ∈ V
rankelun.2 𝐵 ∈ V
rankelun.3 𝐶 ∈ V
rankelun.4 𝐷 ∈ V
Assertion
Ref Expression
rankelop (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘⟨𝐴, 𝐵⟩) ∈ (rank‘⟨𝐶, 𝐷⟩))

Proof of Theorem rankelop
StepHypRef Expression
1 rankelun.1 . . . 4 𝐴 ∈ V
2 rankelun.2 . . . 4 𝐵 ∈ V
3 rankelun.3 . . . 4 𝐶 ∈ V
4 rankelun.4 . . . 4 𝐷 ∈ V
51, 2, 3, 4rankelpr 9286 . . 3 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))
6 rankon 9208 . . . . 5 (rank‘{𝐶, 𝐷}) ∈ On
76onordi 6263 . . . 4 Ord (rank‘{𝐶, 𝐷})
8 ordsucelsuc 7517 . . . 4 (Ord (rank‘{𝐶, 𝐷}) → ((rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}) ↔ suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷})))
97, 8ax-mp 5 . . 3 ((rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}) ↔ suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷}))
105, 9sylib 221 . 2 (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷}))
111, 2rankop 9271 . . 3 (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))
121, 2rankpr 9270 . . . 4 (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))
13 suceq 6224 . . . 4 ((rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) → suc (rank‘{𝐴, 𝐵}) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
1412, 13ax-mp 5 . . 3 suc (rank‘{𝐴, 𝐵}) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))
1511, 14eqtr4i 2824 . 2 (rank‘⟨𝐴, 𝐵⟩) = suc (rank‘{𝐴, 𝐵})
163, 4rankop 9271 . . 3 (rank‘⟨𝐶, 𝐷⟩) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷))
173, 4rankpr 9270 . . . 4 (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷))
18 suceq 6224 . . . 4 ((rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷)) → suc (rank‘{𝐶, 𝐷}) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷)))
1917, 18ax-mp 5 . . 3 suc (rank‘{𝐶, 𝐷}) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷))
2016, 19eqtr4i 2824 . 2 (rank‘⟨𝐶, 𝐷⟩) = suc (rank‘{𝐶, 𝐷})
2110, 15, 203eltr4g 2907 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   = wceq 1538  wcel 2111  Vcvv 3441  cun 3879  {cpr 4527  cop 4531  Ord word 6158  suc csuc 6161  cfv 6324  rankcrnk 9176
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-reg 9040  ax-inf2 9088
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-r1 9177  df-rank 9178
This theorem is referenced by: (None)
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