Theorem r1ord3 5768

Description: Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478.

Assertion

Ref	Expression
r1ord3	|- ((A e. On /\ B e. On) -> (A C_ B -> (R1` A) C_ (R1` B)))

Proof of Theorem r1ord3

Step	Hyp	Ref	Expression
1		onsseleq 3704	. 2 |- ((A e. On /\ B e. On) -> (A C_ B <-> (A e. B \/ A = B)))
2		r1ord2 5767	. . . 4 |- (B e. On -> (A e. B -> (R1` A) C_ (R1` B)))
3	2	adantl 424	. . 3 |- ((A e. On /\ B e. On) -> (A e. B -> (R1` A) C_ (R1` B)))
4		fveq2 4681	. . . . 5 |- (A = B -> (R1` A) = (R1` B))
5		eqimss 2665	. . . . 5 |- ((R1` A) = (R1` B) -> (R1` A) C_ (R1` B))
6	4, 5	syl 12	. . . 4 |- (A = B -> (R1` A) C_ (R1` B))
7	6	a1i 8	. . 3 |- ((A e. On /\ B e. On) -> (A = B -> (R1` A) C_ (R1` B)))
8	3, 7	jaod 469	. 2 |- ((A e. On /\ B e. On) -> ((A e. B \/ A = B) -> (R1` A) C_ (R1` B)))
9	1, 8	sylbid 220	1 |- ((A e. On /\ B e. On) -> (A C_ B -> (R1` A) C_ (R1` B)))

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  Oncon0 3657  ` cfv 3998  R1cr1 5748

This theorem is referenced by:  r1val1 5769  rankr1lem 5784  ssrankr1 5787  rankel 5791  rankval3 5792  bndrank 5793  r1pwcl 5798  rankr1id 5808  rankr1b 5810  rankval4 5813

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750

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