Theorem rankelun 5818

Description: Rank membership is inherited by union.

Hypotheses

Ref	Expression
rankelun.1	|- A e. _V
rankelun.2	|- B e. _V
rankelun.3	|- C e. _V
rankelun.4	|- D e. _V

Assertion

Ref	Expression
rankelun	|- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` (A u. B)) e. (rank` (C u. D)))

Proof of Theorem rankelun

Step	Hyp	Ref	Expression
1		elun1 2771	. . . . . . . 8 |- (A e. (R1` (rank` C)) -> A e. ((R1` (rank` C)) u. (R1` (rank` D))))
2		elun2 2772	. . . . . . . 8 |- (B e. (R1` (rank` D)) -> B e. ((R1` (rank` C)) u. (R1` (rank` D))))
3	1, 2	anim12i 360	. . . . . . 7 |- ((A e. (R1` (rank` C)) /\ B e. (R1` (rank` D))) -> (A e. ((R1` (rank` C)) u. (R1` (rank` D))) /\ B e. ((R1` (rank` C)) u. (R1` (rank` D)))))
4		rankelun.1	. . . . . . . 8 |- A e. _V
5		rankelun.2	. . . . . . . 8 |- B e. _V
6	4, 5	prss 3138	. . . . . . 7 |- ((A e. ((R1` (rank` C)) u. (R1` (rank` D))) /\ B e. ((R1` (rank` C)) u. (R1` (rank` D)))) <-> {A, B} C_ ((R1` (rank` C)) u. (R1` (rank` D))))
7	3, 6	sylib 215	. . . . . 6 |- ((A e. (R1` (rank` C)) /\ B e. (R1` (rank` D))) -> {A, B} C_ ((R1` (rank` C)) u. (R1` (rank` D))))
8		fvex 4689	. . . . . . . 8 |- (R1` (rank` C)) e. _V
9		fvex 4689	. . . . . . . 8 |- (R1` (rank` D)) e. _V
10	8, 9	unex 3796	. . . . . . 7 |- ((R1` (rank` C)) u. (R1` (rank` D))) e. _V
11	10	rankss 5799	. . . . . 6 |- ({A, B} C_ ((R1` (rank` C)) u. (R1` (rank` D))) -> (rank` {A, B}) C_ (rank` ((R1` (rank` C)) u. (R1` (rank` D)))))
12	7, 11	syl 12	. . . . 5 |- ((A e. (R1` (rank` C)) /\ B e. (R1` (rank` D))) -> (rank` {A, B}) C_ (rank` ((R1` (rank` C)) u. (R1` (rank` D)))))
13	4, 5	rankpr 5803	. . . . . 6 |- (rank` {A, B}) = suc ((rank` A) u. (rank` B))
14	4, 5	rankun 5802	. . . . . . 7 |- (rank` (A u. B)) = ((rank` A) u. (rank` B))
15		suceq 3729	. . . . . . 7 |- ((rank` (A u. B)) = ((rank` A) u. (rank` B)) -> suc (rank` (A u. B)) = suc ((rank` A) u. (rank` B)))
16	14, 15	ax-mp 7	. . . . . 6 |- suc (rank` (A u. B)) = suc ((rank` A) u. (rank` B))
17	13, 16	eqtr4i 1911	. . . . 5 |- (rank` {A, B}) = suc (rank` (A u. B))
18	12, 17	syl5ssr 2662	. . . 4 |- ((A e. (R1` (rank` C)) /\ B e. (R1` (rank` D))) -> suc (rank` (A u. B)) C_ (rank` ((R1` (rank` C)) u. (R1` (rank` D)))))
19		fvex 4689	. . . . 5 |- (rank` (A u. B)) e. _V
20		sucssel 3763	. . . . 5 |- ((rank` (A u. B)) e. _V -> (suc (rank` (A u. B)) C_ (rank` ((R1` (rank` C)) u. (R1` (rank` D)))) -> (rank` (A u. B)) e. (rank` ((R1` (rank` C)) u. (R1` (rank` D))))))
21	19, 20	ax-mp 7	. . . 4 |- (suc (rank` (A u. B)) C_ (rank` ((R1` (rank` C)) u. (R1` (rank` D)))) -> (rank` (A u. B)) e. (rank` ((R1` (rank` C)) u. (R1` (rank` D)))))
22	18, 21	syl 12	. . 3 |- ((A e. (R1` (rank` C)) /\ B e. (R1` (rank` D))) -> (rank` (A u. B)) e. (rank` ((R1` (rank` C)) u. (R1` (rank` D)))))
23		rankon 5782	. . . 4 |- (rank` C) e. On
24	4	rankr1a 5788	. . . 4 |- ((rank` C) e. On -> (A e. (R1` (rank` C)) <-> (rank` A) e. (rank` C)))
25	23, 24	ax-mp 7	. . 3 |- (A e. (R1` (rank` C)) <-> (rank` A) e. (rank` C))
26		rankon 5782	. . . 4 |- (rank` D) e. On
27	5	rankr1a 5788	. . . 4 |- ((rank` D) e. On -> (B e. (R1` (rank` D)) <-> (rank` B) e. (rank` D)))
28	26, 27	ax-mp 7	. . 3 |- (B e. (R1` (rank` D)) <-> (rank` B) e. (rank` D))
29	22, 25, 28	syl2anbr 505	. 2 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` (A u. B)) e. (rank` ((R1` (rank` C)) u. (R1` (rank` D)))))
30		rankr1id 5808	. . . . 5 |- ((rank` C) e. On <-> (rank` (R1` (rank` C))) = (rank` C))
31	23, 30	mpbi 206	. . . 4 |- (rank` (R1` (rank` C))) = (rank` C)
32		rankr1id 5808	. . . . 5 |- ((rank` D) e. On <-> (rank` (R1` (rank` D))) = (rank` D))
33	26, 32	mpbi 206	. . . 4 |- (rank` (R1` (rank` D))) = (rank` D)
34	31, 33	uneq12i 2753	. . 3 |- ((rank` (R1` (rank` C))) u. (rank` (R1` (rank` D)))) = ((rank` C) u. (rank` D))
35	8, 9	rankun 5802	. . 3 |- (rank` ((R1` (rank` C)) u. (R1` (rank` D)))) = ((rank` (R1` (rank` C))) u. (rank` (R1` (rank` D))))
36		rankelun.3	. . . 4 |- C e. _V
37		rankelun.4	. . . 4 |- D e. _V
38	36, 37	rankun 5802	. . 3 |- (rank` (C u. D)) = ((rank` C) u. (rank` D))
39	34, 35, 38	3eqtr4i 1921	. 2 |- (rank` ((R1` (rank` C)) u. (R1` (rank` D)))) = (rank` (C u. D))
40	29, 39	syl6eleq 1981	1 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` (A u. B)) e. (rank` (C u. D)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   u. cun 2591   C_ wss 2593  {cpr 3045  Oncon0 3657  suc csuc 3659  ` cfv 3998  R1cr1 5748  rankcrnk 5749

This theorem is referenced by:  rankelpr 5819  rankxplim 5823

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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