Theorem rankelop 9297

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

Step	Hyp	Ref	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
5	1, 2, 3, 4	rankelpr 9296	. . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))
6		rankon 9218	. . . . 5 ⊢ (rank‘{𝐶, 𝐷}) ∈ On
7	6	onordi 6289	. . . 4 ⊢ Ord (rank‘{𝐶, 𝐷})
8		ordsucelsuc 7531	. . . 4 ⊢ (Ord (rank‘{𝐶, 𝐷}) → ((rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}) ↔ suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷})))
9	7, 8	ax-mp 5	. . 3 ⊢ ((rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}) ↔ suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷}))
10	5, 9	sylib 220	. 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷}))
11	1, 2	rankop 9281	. . 3 ⊢ (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))
12	1, 2	rankpr 9280	. . . 4 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))
13		suceq 6250	. . . 4 ⊢ ((rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) → suc (rank‘{𝐴, 𝐵}) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
14	12, 13	ax-mp 5	. . 3 ⊢ suc (rank‘{𝐴, 𝐵}) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))
15	11, 14	eqtr4i 2847	. 2 ⊢ (rank‘⟨𝐴, 𝐵⟩) = suc (rank‘{𝐴, 𝐵})
16	3, 4	rankop 9281	. . 3 ⊢ (rank‘⟨𝐶, 𝐷⟩) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷))
17	3, 4	rankpr 9280	. . . 4 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷))
18		suceq 6250	. . . 4 ⊢ ((rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷)) → suc (rank‘{𝐶, 𝐷}) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷)))
19	17, 18	ax-mp 5	. . 3 ⊢ suc (rank‘{𝐶, 𝐷}) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷))
20	16, 19	eqtr4i 2847	. 2 ⊢ (rank‘⟨𝐶, 𝐷⟩) = suc (rank‘{𝐶, 𝐷})
21	10, 15, 20	3eltr4g 2930	1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘⟨𝐴, 𝐵⟩) ∈ (rank‘⟨𝐶, 𝐷⟩))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∪ cun 3933  {cpr 4562  ⟨cop 4566  Ord word 6184  suc csuc 6187  ‘cfv 6349  rankcrnk 9186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-reg 9050  ax-inf2 9098

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-r1 9187  df-rank 9188

This theorem is referenced by: (None)