Theorem rankss 4698

Description: The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80.

Hypothesis

Ref	Expression
rankss.1	|- B e. V

Assertion

Ref	Expression
rankss	|- (A (_ B -> (rank` A) (_ (rank` B))

Proof of Theorem rankss

Step	Hyp	Ref	Expression
1		rankss.1	. . . 4 |- B e. V
2	1	pwex 2751	. . 3 |- P~B e. V
3	2	rankel 4690	. 2 |- (A e. P~B -> (rank` A) e. (rank` P~B))
4	1	elpw2 2733	. 2 |- (A e. P~B <-> A (_ B)
5	1	rankpw 4694	. . . 4 |- (rank` P~B) = suc (rank` B)
6	5	eleq2i 1541	. . 3 |- ((rank` A) e. (rank` P~B) <-> (rank` A) e. suc (rank` B))
7		rankon 4681	. . . 4 |- (rank` A) e. On
8		rankon 4681	. . . 4 |- (rank` B) e. On
9		onsssuc 3064	. . . 4 |- (((rank` A) e. On /\ (rank` B) e. On) -> ((rank` A) (_ (rank` B) <-> (rank` A) e. suc (rank` B)))
10	7, 8, 9	mp2an 699	. . 3 |- ((rank` A) (_ (rank` B) <-> (rank` A) e. suc (rank` B))
11	6, 10	bitr4 176	. 2 |- ((rank` A) e. (rank` P~B) <-> (rank` A) (_ (rank` B))
12	3, 4, 11	3imtr3 218	1 |- (A (_ B -> (rank` A) (_ (rank` B))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 960  Vcvv 1814   (_ wss 2050  P~cpw 2405  Oncon0 2954  suc csuc 2956  ` cfv 3188  rankcrnk 4652

This theorem is referenced by:  rankuni2 4700  rankun 4701  rankr1id 4707  rankuni 4708  rankr1b 4709  rankval4 4712  rankc2 4716  rankelun 4717  rankxpu 4721  rankxplim 4722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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