Theorem rankpr 4702

Description: The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207.

Hypotheses

Ref	Expression
rankun.1	⊢ A ∈ V
rankun.2	⊢ B ∈ V

Assertion

Ref	Expression
rankpr	⊢ (rank ‘{A, B}) = suc ((rank ‘A) ∪ (rank ‘B))

Proof of Theorem rankpr

Step	Hyp	Ref	Expression
1		df-pr 2417	. . . 4 ⊢ {A, B} = ({A} ∪ {B})
2	1	fveq2i 3733	. . 3 ⊢ (rank ‘{A, B}) = (rank ‘({A} ∪ {B}))
3		snex 2756	. . . 4 ⊢ {A} ∈ V
4		snex 2756	. . . 4 ⊢ {B} ∈ V
5	3, 4	rankun 4701	. . 3 ⊢ (rank ‘({A} ∪ {B})) = ((rank ‘{A}) ∪ (rank ‘{B}))
6		rankun.1	. . . . 5 ⊢ A ∈ V
7	6	ranksn 4699	. . . 4 ⊢ (rank ‘{A}) = suc (rank ‘A)
8		rankun.2	. . . . 5 ⊢ B ∈ V
9	8	ranksn 4699	. . . 4 ⊢ (rank ‘{B}) = suc (rank ‘B)
10	7, 9	uneq12i 2185	. . 3 ⊢ ((rank ‘{A}) ∪ (rank ‘{B})) = (suc (rank ‘A) ∪ suc (rank ‘B))
11	2, 5, 10	3eqtr 1502	. 2 ⊢ (rank ‘{A, B}) = (suc (rank ‘A) ∪ suc (rank ‘B))
12		rankon 4681	. . . 4 ⊢ (rank ‘A) ∈ On
13	12	onord 3101	. . 3 ⊢ Ord (rank ‘A)
14		rankon 4681	. . . 4 ⊢ (rank ‘B) ∈ On
15	14	onord 3101	. . 3 ⊢ Ord (rank ‘B)
16		ordsucun 3088	. . 3 ⊢ ((Ord (rank ‘A) ⋀ Ord (rank ‘B)) → suc ((rank ‘A) ∪ (rank ‘B)) = (suc (rank ‘A) ∪ suc (rank ‘B)))
17	13, 15, 16	mp2an 699	. 2 ⊢ suc ((rank ‘A) ∪ (rank ‘B)) = (suc (rank ‘A) ∪ suc (rank ‘B))
18	11, 17	eqtr4 1501	1 ⊢ (rank ‘{A, B}) = suc ((rank ‘A) ∪ (rank ‘B))

Syntax hints:   = wceq 958   ∈ wcel 960  Vcvv 1814   ∪ cun 2048  {csn 2413  {cpr 2414  Ord word 2953  suc csuc 2956   ‘cfv 3188  rankcrnk 4652

This theorem is referenced by:  rankop 4703  rankelun 4717  rankelpr 4718  rankelop 4719

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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