Theorem rankelun 4707

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 2197	. . . . . . . 8 |- (A e. (R1` (rank` C)) -> A e. ((R1` (rank` C)) u. (R1` (rank` D))))
2		elun2 2198	. . . . . . . 8 |- (B e. (R1` (rank` D)) -> B e. ((R1` (rank` C)) u. (R1` (rank` D))))
3	1, 2	anim12i 333	. . . . . . 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 2471	. . . . . . 7 |- ((A e. ((R1` (rank` C)) u. (R1` (rank` D))) /\ B e. ((R1` (rank` C)) u. (R1` (rank` D)))) <-> {A, B} (_ ((R1` (rank` C)) u. (R1` (rank` D))))
7	3, 6	sylib 198	. . . . . 6 |- ((A e. (R1` (rank` C)) /\ B e. (R1` (rank` D))) -> {A, B} (_ ((R1` (rank` C)) u. (R1` (rank` D))))
8		fvex 3732	. . . . . . . 8 |- (R1` (rank` C)) e. V
9		fvex 3732	. . . . . . . 8 |- (R1` (rank` D)) e. V
10	8, 9	unex 2872	. . . . . . 7 |- ((R1` (rank` C)) u. (R1` (rank` D))) e. V
11	10	rankss 4688	. . . . . 6 |- ({A, B} (_ ((R1` (rank` C)) u. (R1` (rank` D))) -> (rank` {A, B}) (_ (rank` ((R1` (rank` C)) u. (R1` (rank` D)))))
12	7, 11	syl 10	. . . . 5 |- ((A e. (R1` (rank` C)) /\ B e. (R1` (rank` D))) -> (rank` {A, B}) (_ (rank` ((R1` (rank` C)) u. (R1` (rank` D)))))
13	4, 5	rankpr 4692	. . . . . 6 |- (rank` {A, B}) = suc ((rank` A) u. (rank` B))
14	4, 5	rankun 4691	. . . . . . 7 |- (rank` (A u. B)) = ((rank` A) u. (rank` B))
15		suceq 3034	. . . . . . 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	eqtr4 1498	. . . . 5 |- (rank` {A, B}) = suc (rank` (A u. B))
18	12, 17	syl5ssr 2106	. . . 4 |- ((A e. (R1` (rank` C)) /\ B e. (R1` (rank` D))) -> suc (rank` (A u. B)) (_ (rank` ((R1` (rank` C)) u. (R1` (rank` D)))))
19		fvex 3732	. . . . 5 |- (rank` (A u. B)) e. V
20		sucssel 3070	. . . . 5 |- ((rank` (A u. B)) e. V -> (suc (rank` (A u. B)) (_ (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)) (_ (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 10	. . 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 4671	. . . 4 |- (rank` C) e. On
24	4	rankr1a 4677	. . . 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 4671	. . . 4 |- (rank` D) e. On
27	5	rankr1a 4677	. . . 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 456	. 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 4697	. . . . 5 |- ((rank` C) e. On <-> (rank` (R1` (rank` C))) = (rank` C))
31	23, 30	mpbi 189	. . . 4 |- (rank` (R1` (rank` C))) = (rank` C)
32		rankr1id 4697	. . . . 5 |- ((rank` D) e. On <-> (rank` (R1` (rank` D))) = (rank` D))
33	26, 32	mpbi 189	. . . 4 |- (rank` (R1` (rank` D))) = (rank` D)
34	31, 33	uneq12i 2182	. . 3 |- ((rank` (R1` (rank` C))) u. (rank` (R1` (rank` D)))) = ((rank` C) u. (rank` D))
35	8, 9	rankun 4691	. . 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 4691	. . 3 |- (rank` (C u. D)) = ((rank` C) u. (rank` D))
39	34, 35, 38	3eqtr4 1505	. 2 |- (rank` ((R1` (rank` C)) u. (R1` (rank` D)))) = (rank` (C u. D))
40	29, 39	syl6eleq 1558	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 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045   (_ wss 2047  {cpr 2410  Oncon0 2948  suc csuc 2950  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem is referenced by:  rankelpr 4708  rankxplim 4712

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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