Theorem rankun 4671

Description: The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112.

Hypotheses

Ref	Expression
rankun.1	|- A e. V
rankun.2	|- B e. V

Assertion

Ref	Expression
rankun	|- (rank` (A u. B)) = ((rank` A) u. (rank` B))

Proof of Theorem rankun

Step	Hyp	Ref	Expression
1		rankun.1	. . . . . . . 8 |- A e. V
2		rankun.2	. . . . . . . 8 |- B e. V
3	1, 2	unex 2867	. . . . . . 7 |- (A u. B) e. V
4	3	rankval3 4661	. . . . . 6 |- (rank` (A u. B)) = |^|{y e. On | A.z e. (A u. B)(rank` z) e. y}
5	4	eleq2i 1535	. . . . 5 |- (x e. (rank` (A u. B)) <-> x e. |^|{y e. On | A.z e. (A u. B)(rank` z) e. y})
6		visset 1809	. . . . . 6 |- x e. V
7	6	elintrab 2540	. . . . 5 |- (x e. |^|{y e. On | A.z e. (A u. B)(rank` z) e. y} <-> A.y e. On (A.z e. (A u. B)(rank` z) e. y -> x e. y))
8	5, 7	bitr 173	. . . 4 |- (x e. (rank` (A u. B)) <-> A.y e. On (A.z e. (A u. B)(rank` z) e. y -> x e. y))
9		elun 2169	. . . . . . 7 |- (z e. (A u. B) <-> (z e. A \/ z e. B))
10	1	rankel 4660	. . . . . . . . 9 |- (z e. A -> (rank` z) e. (rank` A))
11		elun1 2193	. . . . . . . . 9 |- ((rank` z) e. (rank` A) -> (rank` z) e. ((rank` A) u. (rank` B)))
12	10, 11	syl 10	. . . . . . . 8 |- (z e. A -> (rank` z) e. ((rank` A) u. (rank` B)))
13	2	rankel 4660	. . . . . . . . 9 |- (z e. B -> (rank` z) e. (rank` B))
14		elun2 2194	. . . . . . . . 9 |- ((rank` z) e. (rank` B) -> (rank` z) e. ((rank` A) u. (rank` B)))
15	13, 14	syl 10	. . . . . . . 8 |- (z e. B -> (rank` z) e. ((rank` A) u. (rank` B)))
16	12, 15	jaoi 341	. . . . . . 7 |- ((z e. A \/ z e. B) -> (rank` z) e. ((rank` A) u. (rank` B)))
17	9, 16	sylbi 199	. . . . . 6 |- (z e. (A u. B) -> (rank` z) e. ((rank` A) u. (rank` B)))
18	17	rgen 1695	. . . . 5 |- A.z e. (A u. B)(rank` z) e. ((rank` A) u. (rank` B))
19		rankon 4651	. . . . . . 7 |- (rank` A) e. On
20		rankon 4651	. . . . . . 7 |- (rank` B) e. On
21	19, 20	onun 3105	. . . . . 6 |- ((rank` A) u. (rank` B)) e. On
22		eleq2 1532	. . . . . . . . 9 |- (y = ((rank` A) u. (rank` B)) -> ((rank` z) e. y <-> (rank` z) e. ((rank` A) u. (rank` B))))
23	22	ralbidv 1660	. . . . . . . 8 |- (y = ((rank` A) u. (rank` B)) -> (A.z e. (A u. B)(rank` z) e. y <-> A.z e. (A u. B)(rank` z) e. ((rank` A) u. (rank` B))))
24		eleq2 1532	. . . . . . . 8 |- (y = ((rank` A) u. (rank` B)) -> (x e. y <-> x e. ((rank` A) u. (rank` B))))
25	23, 24	imbi12d 625	. . . . . . 7 |- (y = ((rank` A) u. (rank` B)) -> ((A.z e. (A u. B)(rank` z) e. y -> x e. y) <-> (A.z e. (A u. B)(rank` z) e. ((rank` A) u. (rank` B)) -> x e. ((rank` A) u. (rank` B)))))
26	25	rcla4v 1869	. . . . . 6 |- (((rank` A) u. (rank` B)) e. On -> (A.y e. On (A.z e. (A u. B)(rank` z) e. y -> x e. y) -> (A.z e. (A u. B)(rank` z) e. ((rank` A) u. (rank` B)) -> x e. ((rank` A) u. (rank` B)))))
27	21, 26	ax-mp 7	. . . . 5 |- (A.y e. On (A.z e. (A u. B)(rank` z) e. y -> x e. y) -> (A.z e. (A u. B)(rank` z) e. ((rank` A) u. (rank` B)) -> x e. ((rank` A) u. (rank` B))))
28	18, 27	mpi 44	. . . 4 |- (A.y e. On (A.z e. (A u. B)(rank` z) e. y -> x e. y) -> x e. ((rank` A) u. (rank` B)))
29	8, 28	sylbi 199	. . 3 |- (x e. (rank` (A u. B)) -> x e. ((rank` A) u. (rank` B)))
30	29	ssriv 2065	. 2 |- (rank` (A u. B)) (_ ((rank` A) u. (rank` B))
31		ssun1 2189	. . . 4 |- A (_ (A u. B)
32	3	rankss 4668	. . . 4 |- (A (_ (A u. B) -> (rank` A) (_ (rank` (A u. B)))
33	31, 32	ax-mp 7	. . 3 |- (rank` A) (_ (rank` (A u. B))
34		ssun2 2190	. . . 4 |- B (_ (A u. B)
35	3	rankss 4668	. . . 4 |- (B (_ (A u. B) -> (rank` B) (_ (rank` (A u. B)))
36	34, 35	ax-mp 7	. . 3 |- (rank` B) (_ (rank` (A u. B))
37	33, 36	unssi 2201	. 2 |- ((rank` A) u. (rank` B)) (_ (rank` (A u. B))
38	30, 37	eqssi 2074	1 |- (rank` (A u. B)) = ((rank` A) u. (rank` B))

Syntax hints:   -> wi 3   \/ wo 222   = wceq 954   e. wcel 956  A.wral 1642  {crab 1645  Vcvv 1807   u. cun 2041   (_ wss 2043  |^|cint 2528  Oncon0 2943  ` cfv 3177  rankcrnk 4622

This theorem is referenced by:  rankpr 4672  rankop 4673  ranksuc 4680  rankelun 4687  rankelpr 4688

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623  df-rank 4624

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