Theorem rankun 5802

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 3796	. . . . . . 7 |- (A u. B) e. _V
4	3	rankval3 5792	. . . . . 6 |- (rank` (A u. B)) = |^|{y e. On | A.z e. (A u. B)(rank` z) e. y}
5	4	eleq2i 1961	. . . . 5 |- (x e. (rank` (A u. B)) <-> x e. |^|{y e. On | A.z e. (A u. B)(rank` z) e. y})
6		visset 2295	. . . . . 6 |- x e. _V
7	6	elintrab 3228	. . . . 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	bitri 190	. . . 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 2741	. . . . . . 7 |- (z e. (A u. B) <-> (z e. A \/ z e. B))
10	1	rankel 5791	. . . . . . . . 9 |- (z e. A -> (rank` z) e. (rank` A))
11		elun1 2771	. . . . . . . . 9 |- ((rank` z) e. (rank` A) -> (rank` z) e. ((rank` A) u. (rank` B)))
12	10, 11	syl 12	. . . . . . . 8 |- (z e. A -> (rank` z) e. ((rank` A) u. (rank` B)))
13	2	rankel 5791	. . . . . . . . 9 |- (z e. B -> (rank` z) e. (rank` B))
14		elun2 2772	. . . . . . . . 9 |- ((rank` z) e. (rank` B) -> (rank` z) e. ((rank` A) u. (rank` B)))
15	13, 14	syl 12	. . . . . . . 8 |- (z e. B -> (rank` z) e. ((rank` A) u. (rank` B)))
16	12, 15	jaoi 368	. . . . . . 7 |- ((z e. A \/ z e. B) -> (rank` z) e. ((rank` A) u. (rank` B)))
17	9, 16	sylbi 216	. . . . . 6 |- (z e. (A u. B) -> (rank` z) e. ((rank` A) u. (rank` B)))
18	17	rgen 2159	. . . . 5 |- A.z e. (A u. B)(rank` z) e. ((rank` A) u. (rank` B))
19		rankon 5782	. . . . . . 7 |- (rank` A) e. On
20		rankon 5782	. . . . . . 7 |- (rank` B) e. On
21	19, 20	onun2i 3785	. . . . . 6 |- ((rank` A) u. (rank` B)) e. On
22		eleq2 1958	. . . . . . . . 9 |- (y = ((rank` A) u. (rank` B)) -> ((rank` z) e. y <-> (rank` z) e. ((rank` A) u. (rank` B))))
23	22	ralbidv 2123	. . . . . . . 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 1958	. . . . . . . 8 |- (y = ((rank` A) u. (rank` B)) -> (x e. y <-> x e. ((rank` A) u. (rank` B))))
25	23, 24	imbi12d 688	. . . . . . 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 2376	. . . . . 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 55	. . . 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 216	. . 3 |- (x e. (rank` (A u. B)) -> x e. ((rank` A) u. (rank` B)))
30	29	ssriv 2621	. 2 |- (rank` (A u. B)) C_ ((rank` A) u. (rank` B))
31		ssun1 2767	. . . 4 |- A C_ (A u. B)
32	3	rankss 5799	. . . 4 |- (A C_ (A u. B) -> (rank` A) C_ (rank` (A u. B)))
33	31, 32	ax-mp 7	. . 3 |- (rank` A) C_ (rank` (A u. B))
34		ssun2 2768	. . . 4 |- B C_ (A u. B)
35	3	rankss 5799	. . . 4 |- (B C_ (A u. B) -> (rank` B) C_ (rank` (A u. B)))
36	34, 35	ax-mp 7	. . 3 |- (rank` B) C_ (rank` (A u. B))
37	33, 36	unssi 2781	. 2 |- ((rank` A) u. (rank` B)) C_ (rank` (A u. B))
38	30, 37	eqssi 2632	1 |- (rank` (A u. B)) = ((rank` A) u. (rank` B))

Syntax hints:   -> wi 3   \/ wo 239   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292   u. cun 2591   C_ wss 2593  |^|cint 3214  Oncon0 3657  ` cfv 3998  rankcrnk 5749

This theorem is referenced by:  rankpr 5803  rankop 5804  ranksuc 5811  rankelun 5818  rankelpr 5819

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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