Theorem rankelop 4719

Description: Rank membership is inherited by ordered pairs.

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	. . . . 5 ⊢ A ∈ V
2		rankelun.2	. . . . 5 ⊢ B ∈ V
3		rankelun.3	. . . . 5 ⊢ C ∈ V
4		rankelun.4	. . . . 5 ⊢ D ∈ V
5	1, 2, 3, 4	rankelpr 4718	. . . 4 ⊢ (((rank ‘A) ∈ (rank ‘C) ⋀ (rank ‘B) ∈ (rank ‘D)) → (rank ‘{A, B}) ∈ (rank ‘{C, D}))
6		rankon 4681	. . . . . 6 ⊢ (rank ‘{C, D}) ∈ On
7	6	onord 3101	. . . . 5 ⊢ Ord (rank ‘{C, D})
8		ordsucelsuc 3079	. . . . 5 ⊢ (Ord (rank ‘{C, D}) → ((rank ‘{A, B}) ∈ (rank ‘{C, D}) ↔ suc (rank ‘{A, B}) ∈ suc (rank ‘{C, D})))
9	7, 8	ax-mp 7	. . . 4 ⊢ ((rank ‘{A, B}) ∈ (rank ‘{C, D}) ↔ suc (rank ‘{A, B}) ∈ suc (rank ‘{C, D}))
10	5, 9	sylib 198	. . 3 ⊢ (((rank ‘A) ∈ (rank ‘C) ⋀ (rank ‘B) ∈ (rank ‘D)) → suc (rank ‘{A, B}) ∈ suc (rank ‘{C, D}))
11	3, 4	rankpr 4702	. . . . 5 ⊢ (rank ‘{C, D}) = suc ((rank ‘C) ∪ (rank ‘D))
12		suceq 3040	. . . . 5 ⊢ ((rank ‘{C, D}) = suc ((rank ‘C) ∪ (rank ‘D)) → suc (rank ‘{C, D}) = suc suc ((rank ‘C) ∪ (rank ‘D)))
13	11, 12	ax-mp 7	. . . 4 ⊢ suc (rank ‘{C, D}) = suc suc ((rank ‘C) ∪ (rank ‘D))
14	3, 4	rankop 4703	. . . 4 ⊢ (rank ‘⟨C, D⟩) = suc suc ((rank ‘C) ∪ (rank ‘D))
15	13, 14	eqtr4 1501	. . 3 ⊢ suc (rank ‘{C, D}) = (rank ‘⟨C, D⟩)
16	10, 15	syl6eleq 1561	. 2 ⊢ (((rank ‘A) ∈ (rank ‘C) ⋀ (rank ‘B) ∈ (rank ‘D)) → suc (rank ‘{A, B}) ∈ (rank ‘⟨C, D⟩))
17	1, 2	rankop 4703	. . 3 ⊢ (rank ‘⟨A, B⟩) = suc suc ((rank ‘A) ∪ (rank ‘B))
18	1, 2	rankpr 4702	. . . 4 ⊢ (rank ‘{A, B}) = suc ((rank ‘A) ∪ (rank ‘B))
19		suceq 3040	. . . 4 ⊢ ((rank ‘{A, B}) = suc ((rank ‘A) ∪ (rank ‘B)) → suc (rank ‘{A, B}) = suc suc ((rank ‘A) ∪ (rank ‘B)))
20	18, 19	ax-mp 7	. . 3 ⊢ suc (rank ‘{A, B}) = suc suc ((rank ‘A) ∪ (rank ‘B))
21	17, 20	eqtr4 1501	. 2 ⊢ (rank ‘⟨A, B⟩) = suc (rank ‘{A, B})
22	16, 21	syl5eqel 1555	1 ⊢ (((rank ‘A) ∈ (rank ‘C) ⋀ (rank ‘B) ∈ (rank ‘D)) → (rank ‘⟨A, B⟩) ∈ (rank ‘⟨C, D⟩))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  Vcvv 1814   ∪ cun 2048  {cpr 2414  ⟨cop 2415  Ord word 2953  suc csuc 2956   ‘cfv 3188  rankcrnk 4652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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