Theorem rankelpr 4688

Description: Rank membership is inherited by unordered 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
rankelpr	|- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` {A, B}) e. (rank` {C, D}))

Proof of Theorem rankelpr

Step	Hyp	Ref	Expression
1		rankelun.1	. . . . . . 7 |- A e. V
2		rankelun.2	. . . . . . 7 |- B e. V
3		rankelun.3	. . . . . . 7 |- C e. V
4		rankelun.4	. . . . . . 7 |- D e. V
5	1, 2, 3, 4	rankelun 4687	. . . . . 6 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` (A u. B)) e. (rank` (C u. D)))
6	3, 4	rankun 4671	. . . . . 6 |- (rank` (C u. D)) = ((rank` C) u. (rank` D))
7	5, 6	syl6eleq 1555	. . . . 5 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` (A u. B)) e. ((rank` C) u. (rank` D)))
8	1, 2	rankun 4671	. . . . 5 |- (rank` (A u. B)) = ((rank` A) u. (rank` B))
9	7, 8	syl5eqelr 1550	. . . 4 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> ((rank` A) u. (rank` B)) e. ((rank` C) u. (rank` D)))
10		rankon 4651	. . . . . . 7 |- (rank` C) e. On
11		rankon 4651	. . . . . . 7 |- (rank` D) e. On
12	10, 11	onun 3105	. . . . . 6 |- ((rank` C) u. (rank` D)) e. On
13	12	onord 3090	. . . . 5 |- Ord ((rank` C) u. (rank` D))
14		ordsucelsuc 3068	. . . . 5 |- (Ord ((rank` C) u. (rank` D)) -> (((rank` A) u. (rank` B)) e. ((rank` C) u. (rank` D)) <-> suc ((rank` A) u. (rank` B)) e. suc ((rank` C) u. (rank` D))))
15	13, 14	ax-mp 7	. . . 4 |- (((rank` A) u. (rank` B)) e. ((rank` C) u. (rank` D)) <-> suc ((rank` A) u. (rank` B)) e. suc ((rank` C) u. (rank` D)))
16	9, 15	sylib 198	. . 3 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> suc ((rank` A) u. (rank` B)) e. suc ((rank` C) u. (rank` D)))
17	3, 4	rankpr 4672	. . 3 |- (rank` {C, D}) = suc ((rank` C) u. (rank` D))
18	16, 17	syl6eleqr 1556	. 2 |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> suc ((rank` A) u. (rank` B)) e. (rank` {C, D}))
19	1, 2	rankpr 4672	. 2 |- (rank` {A, B}) = suc ((rank` A) u. (rank` B))
20	18, 19	syl5eqel 1549	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   e. wcel 956  Vcvv 1807   u. cun 2041  {cpr 2406  Ord word 2942  suc csuc 2945  ` cfv 3177  rankcrnk 4622

This theorem is referenced by:  rankelop 4689

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623  df-rank 4624

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