Theorem rankelop 4709

Description: Rank membership is inherited by ordered pairs.

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
rankelop	|- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` <.A, B>.) e. (rank` <.C, D>.))

Proof of Theorem rankelop

Step	Hyp	Ref	Expression
1		rankelun.1	. . . . 5 |- A e. V
2		rankelun.2	. . . . 5 |- B e. V
3		rankelun.3	. . . . 5 |- C e. V
4		rankelun.4	. . . . 5 |- D e. V
5	1, 2, 3, 4	rankelpr 4708	. . . 4 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` {A, B}) e. (rank` {C, D}))
6		rankon 4671	. . . . . 6 |- (rank` {C, D}) e. On
7	6	onord 3095	. . . . 5 |- Ord (rank` {C, D})
8		ordsucelsuc 3073	. . . . 5 |- (Ord (rank` {C, D}) -> ((rank` {A, B}) e. (rank` {C, D}) <-> suc (rank` {A, B}) e. suc (rank` {C, D})))
9	7, 8	ax-mp 7	. . . 4 |- ((rank` {A, B}) e. (rank` {C, D}) <-> suc (rank` {A, B}) e. suc (rank` {C, D}))
10	5, 9	sylib 198	. . 3 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> suc (rank` {A, B}) e. suc (rank` {C, D}))
11	3, 4	rankpr 4692	. . . . 5 |- (rank` {C, D}) = suc ((rank` C) u. (rank` D))
12		suceq 3034	. . . . 5 |- ((rank` {C, D}) = suc ((rank` C) u. (rank` D)) -> suc (rank` {C, D}) = suc suc ((rank` C) u. (rank` D)))
13	11, 12	ax-mp 7	. . . 4 |- suc (rank` {C, D}) = suc suc ((rank` C) u. (rank` D))
14	3, 4	rankop 4693	. . . 4 |- (rank` <.C, D>.) = suc suc ((rank` C) u. (rank` D))
15	13, 14	eqtr4 1498	. . 3 |- suc (rank` {C, D}) = (rank` <.C, D>.)
16	10, 15	syl6eleq 1558	. 2 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> suc (rank` {A, B}) e. (rank` <.C, D>.))
17	1, 2	rankop 4693	. . 3 |- (rank` <.A, B>.) = suc suc ((rank` A) u. (rank` B))
18	1, 2	rankpr 4692	. . . 4 |- (rank` {A, B}) = suc ((rank` A) u. (rank` B))
19		suceq 3034	. . . 4 |- ((rank` {A, B}) = suc ((rank` A) u. (rank` B)) -> suc (rank` {A, B}) = suc suc ((rank` A) u. (rank` B)))
20	18, 19	ax-mp 7	. . 3 |- suc (rank` {A, B}) = suc suc ((rank` A) u. (rank` B))
21	17, 20	eqtr4 1498	. 2 |- (rank` <.A, B>.) = suc (rank` {A, B})
22	16, 21	syl5eqel 1552	1 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` <.A, B>.) e. (rank` <.C, D>.))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045  {cpr 2410  <.cop 2411  Ord word 2947  suc csuc 2950  ` cfv 3182  rankcrnk 4642

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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