Theorem rankxplim2 4713

Description: If the rank of a cross product is a limit ordinal, so is the rank of the union of its arguments.

Hypotheses

Ref	Expression
rankxplim.1	|- A e. V
rankxplim.2	|- B e. V

Assertion

Ref	Expression
rankxplim2	|- (Lim (rank` (A X. B)) -> Lim (rank` (A u. B)))

Proof of Theorem rankxplim2

Step	Hyp	Ref	Expression
1		0ellim 3031	. . 3 |- (Lim (rank` (A X. B)) -> (/) e. (rank` (A X. B)))
2		n0i 2285	. . 3 |- ((/) e. (rank` (A X. B)) -> -. (rank` (A X. B)) = (/))
3		df-ne 1587	. . . . 5 |- ((A X. B) =/= (/) <-> -. (A X. B) = (/))
4		rankxplim.1	. . . . . . . 8 |- A e. V
5		rankxplim.2	. . . . . . . 8 |- B e. V
6	4, 5	xpex 3260	. . . . . . 7 |- (A X. B) e. V
7	6	rankeq0 4696	. . . . . 6 |- ((A X. B) = (/) <-> (rank` (A X. B)) = (/))
8	7	negbii 187	. . . . 5 |- (-. (A X. B) = (/) <-> -. (rank` (A X. B)) = (/))
9	3, 8	bitr2 174	. . . 4 |- (-. (rank` (A X. B)) = (/) <-> (A X. B) =/= (/))
10	9	biimp 151	. . 3 |- (-. (rank` (A X. B)) = (/) -> (A X. B) =/= (/))
11	1, 2, 10	3syl 20	. 2 |- (Lim (rank` (A X. B)) -> (A X. B) =/= (/))
12		unixp 3517	. . . . . 6 |- ((A X. B) =/= (/) -> U.U.(A X. B) = (A u. B))
13	12	fveq2d 3728	. . . . 5 |- ((A X. B) =/= (/) -> (rank` U.U.(A X. B)) = (rank` (A u. B)))
14		rankuni 4698	. . . . . 6 |- (rank` U.U.(A X. B)) = U.(rank` U.(A X. B))
15		rankuni 4698	. . . . . . 7 |- (rank` U.(A X. B)) = U.(rank` (A X. B))
16	15	unieqi 2511	. . . . . 6 |- U.(rank` U.(A X. B)) = U.U.(rank` (A X. B))
17	14, 16	eqtr2 1496	. . . . 5 |- U.U.(rank` (A X. B)) = (rank` U.U.(A X. B))
18	13, 17	syl5eq 1519	. . . 4 |- ((A X. B) =/= (/) -> U.U.(rank` (A X. B)) = (rank` (A u. B)))
19		limeq 2960	. . . 4 |- (U.U.(rank` (A X. B)) = (rank` (A u. B)) -> (Lim U.U.(rank` (A X. B)) <-> Lim (rank` (A u. B))))
20	18, 19	syl 10	. . 3 |- ((A X. B) =/= (/) -> (Lim U.U.(rank` (A X. B)) <-> Lim (rank` (A u. B))))
21		limuni2 3030	. . . 4 |- (Lim (rank` (A X. B)) -> Lim U.(rank` (A X. B)))
22		limuni2 3030	. . . 4 |- (Lim U.(rank` (A X. B)) -> Lim U.U.(rank` (A X. B)))
23	21, 22	syl 10	. . 3 |- (Lim (rank` (A X. B)) -> Lim U.U.(rank` (A X. B)))
24	20, 23	syl5bi 208	. 2 |- ((A X. B) =/= (/) -> (Lim (rank` (A X. B)) -> Lim (rank` (A u. B))))
25	11, 24	mpcom 49	1 |- (Lim (rank` (A X. B)) -> Lim (rank` (A u. B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958   =/= wne 1585  Vcvv 1811   u. cun 2045  (/)c0 2280  U.cuni 2503  Lim wlim 2949   X. cxp 3168  ` cfv 3182  rankcrnk 4642

This theorem is referenced by:  rankxpsuc 4715

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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