Theorem r1ord2 4656

Description: Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77.

Assertion

Ref	Expression
r1ord2	|- (B e. On -> (A e. B -> (R1` A) (_ (R1` B)))

Proof of Theorem r1ord2

Step	Hyp	Ref	Expression
1		r1ord 4655	. 2 |- (B e. On -> (A e. B -> (R1` A) e. (R1` B)))
2		r1tr 4654	. . 3 |- Tr (R1` B)
3		trss 2689	. . 3 |- (Tr (R1` B) -> ((R1` A) e. (R1` B) -> (R1` A) (_ (R1` B)))
4	2, 3	ax-mp 7	. 2 |- ((R1` A) e. (R1` B) -> (R1` A) (_ (R1` B))
5	1, 4	syl6 22	1 |- (B e. On -> (A e. B -> (R1` A) (_ (R1` B)))

Syntax hints:   -> wi 3   e. wcel 958   (_ wss 2047  Tr wtr 2680  Oncon0 2948  ` cfv 3182  R1cr1 4641

This theorem is referenced by:  r1ord3 4657  r1val1 4658  tz9.12lem3 4661  rankwflem 4665  rankr1 4674

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

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-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-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

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