Theorem rankr1lem 4673

Description: Lemma for rankr1 4674.

Hypothesis

Ref	Expression
rankr1lem.1	|- A e. V

Assertion

Ref	Expression
rankr1lem	|- (B e. On -> (-. A e. (R1` B) -> B (_ (rank` A)))

Proof of Theorem rankr1lem

Step	Hyp	Ref	Expression
1		rankon 4671	. . . . . 6 |- (rank` A) e. On
2	1	onsuc 3105	. . . . 5 |- suc (rank` A) e. On
3		ontri1 2981	. . . . 5 |- ((suc (rank` A) e. On /\ B e. On) -> (suc (rank` A) (_ B <-> -. B e. suc (rank` A)))
4	2, 3	mpan 695	. . . 4 |- (B e. On -> (suc (rank` A) (_ B <-> -. B e. suc (rank` A)))
5		rankr1lem.1	. . . . . 6 |- A e. V
6	5	rankid 4672	. . . . 5 |- A e. (R1` suc (rank` A))
7		r1ord3 4657	. . . . . . 7 |- ((suc (rank` A) e. On /\ B e. On) -> (suc (rank` A) (_ B -> (R1` suc (rank` A)) (_ (R1` B)))
8	2, 7	mpan 695	. . . . . 6 |- (B e. On -> (suc (rank` A) (_ B -> (R1` suc (rank` A)) (_ (R1` B)))
9		ssel 2063	. . . . . 6 |- ((R1` suc (rank` A)) (_ (R1` B) -> (A e. (R1` suc (rank` A)) -> A e. (R1` B)))
10	8, 9	syl6 22	. . . . 5 |- (B e. On -> (suc (rank` A) (_ B -> (A e. (R1` suc (rank` A)) -> A e. (R1` B))))
11	6, 10	mpii 45	. . . 4 |- (B e. On -> (suc (rank` A) (_ B -> A e. (R1` B)))
12	4, 11	sylbird 205	. . 3 |- (B e. On -> (-. B e. suc (rank` A) -> A e. (R1` B)))
13	12	con1d 93	. 2 |- (B e. On -> (-. A e. (R1` B) -> B e. suc (rank` A)))
14		onsssuc 3058	. . 3 |- ((B e. On /\ (rank` A) e. On) -> (B (_ (rank` A) <-> B e. suc (rank` A)))
15	1, 14	mpan2 696	. 2 |- (B e. On -> (B (_ (rank` A) <-> B e. suc (rank` A)))
16	13, 15	sylibrd 204	1 |- (B e. On -> (-. A e. (R1` B) -> B (_ (rank` A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   e. wcel 958  Vcvv 1811   (_ wss 2047  Oncon0 2948  suc csuc 2950  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem is referenced by:  rankr1 4674  ssrankr1 4676

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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