Theorem rankun 4691

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 2872	. . . . . . 7 |- (A u. B) e. V
4	3	rankval3 4681	. . . . . 6 |- (rank` (A u. B)) = |^|{y e. On | A.z e. (A u. B)(rank` z) e. y}
5	4	eleq2i 1538	. . . . 5 |- (x e. (rank` (A u. B)) <-> x e. |^|{y e. On | A.z e. (A u. B)(rank` z) e. y})
6		visset 1813	. . . . . 6 |- x e. V
7	6	elintrab 2545	. . . . 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 2173	. . . . . . 7 |- (z e. (A u. B) <-> (z e. A \/ z e. B))
10	1	rankel 4680	. . . . . . . . 9 |- (z e. A -> (rank` z) e. (rank` A))
11		elun1 2197	. . . . . . . . 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 4680	. . . . . . . . 9 |- (z e. B -> (rank` z) e. (rank` B))
14		elun2 2198	. . . . . . . . 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 1698	. . . . 5 |- A.z e. (A u. B)(rank` z) e. ((rank` A) u. (rank` B))
19		rankon 4671	. . . . . . 7 |- (rank` A) e. On
20		rankon 4671	. . . . . . 7 |- (rank` B) e. On
21	19, 20	onun 3110	. . . . . 6 |- ((rank` A) u. (rank` B)) e. On
22		eleq2 1535	. . . . . . . . 9 |- (y = ((rank` A) u. (rank` B)) -> ((rank` z) e. y <-> (rank` z) e. ((rank` A) u. (rank` B))))
23	22	ralbidv 1663	. . . . . . . 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 1535	. . . . . . . 8 |- (y = ((rank` A) u. (rank` B)) -> (x e. y <-> x e. ((rank` A) u. (rank` B))))
25	23, 24	imbi12d 626	. . . . . . 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 1873	. . . . . 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 2069	. 2 |- (rank` (A u. B)) (_ ((rank` A) u. (rank` B))
31		ssun1 2193	. . . 4 |- A (_ (A u. B)
32	3	rankss 4688	. . . 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 2194	. . . 4 |- B (_ (A u. B)
35	3	rankss 4688	. . . 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 2205	. 2 |- ((rank` A) u. (rank` B)) (_ (rank` (A u. B))
38	30, 37	eqssi 2078	1 |- (rank` (A u. B)) = ((rank` A) u. (rank` B))

Syntax hints:   -> wi 3   \/ wo 222   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648  Vcvv 1811   u. cun 2045   (_ wss 2047  |^|cint 2533  Oncon0 2948  ` cfv 3182  rankcrnk 4642

This theorem is referenced by:  rankpr 4692  rankop 4693  ranksuc 4700  rankelun 4707  rankelpr 4708

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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