Theorem rankelop 9632

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 9631	. . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))
6		rankon 9553	. . . . 5 ⊢ (rank‘{𝐶, 𝐷}) ∈ On
7	6	onordi 6371	. . . 4 ⊢ Ord (rank‘{𝐶, 𝐷})
8		ordsucelsuc 7669	. . . 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 217	. 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc (rank‘{𝐴, 𝐵}) ∈ suc (rank‘{𝐶, 𝐷}))
11	1, 2	rankop 9616	. . 3 ⊢ (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))
12	1, 2	rankpr 9615	. . . 4 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))
13		suceq 6331	. . . 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 2769	. 2 ⊢ (rank‘⟨𝐴, 𝐵⟩) = suc (rank‘{𝐴, 𝐵})
16	3, 4	rankop 9616	. . 3 ⊢ (rank‘⟨𝐶, 𝐷⟩) = suc suc ((rank‘𝐶) ∪ (rank‘𝐷))
17	3, 4	rankpr 9615	. . . 4 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷))
18		suceq 6331	. . . 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 2769	. 2 ⊢ (rank‘⟨𝐶, 𝐷⟩) = suc (rank‘{𝐶, 𝐷})
21	10, 15, 20	3eltr4g 2856	1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘⟨𝐴, 𝐵⟩) ∈ (rank‘⟨𝐶, 𝐷⟩))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885  {cpr 4563  ⟨cop 4567  Ord word 6265  suc csuc 6268  ‘cfv 6433  rankcrnk 9521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523

This theorem is referenced by: (None)