Theorem rankxplim2 4693

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 3026	. . 3 |- (Lim (rank` (A X. B)) -> (/) e. (rank` (A X. B)))
2		n0i 2281	. . 3 |- ((/) e. (rank` (A X. B)) -> -. (rank` (A X. B)) = (/))
3		df-ne 1584	. . . . 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 3255	. . . . . . 7 |- (A X. B) e. V
7	6	rankeq0 4676	. . . . . 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 3509	. . . . . 6 |- ((A X. B) =/= (/) -> U.U.(A X. B) = (A u. B))
13	12	fveq2d 3719	. . . . 5 |- ((A X. B) =/= (/) -> (rank` U.U.(A X. B)) = (rank` (A u. B)))
14		rankuni 4678	. . . . . 6 |- (rank` U.U.(A X. B)) = U.(rank` U.(A X. B))
15		rankuni 4678	. . . . . . 7 |- (rank` U.(A X. B)) = U.(rank` (A X. B))
16	15	unieqi 2506	. . . . . 6 |- U.(rank` U.(A X. B)) = U.U.(rank` (A X. B))
17	14, 16	eqtr2 1493	. . . . 5 |- U.U.(rank` (A X. B)) = (rank` U.U.(A X. B))
18	13, 17	syl5eq 1516	. . . 4 |- ((A X. B) =/= (/) -> U.U.(rank` (A X. B)) = (rank` (A u. B)))
19		limeq 2955	. . . 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 3025	. . . 4 |- (Lim (rank` (A X. B)) -> Lim U.(rank` (A X. B)))
22		limuni2 3025	. . . 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 954   e. wcel 956   =/= wne 1582  Vcvv 1807   u. cun 2041  (/)c0 2276  U.cuni 2498  Lim wlim 2944   X. cxp 3163  ` cfv 3177  rankcrnk 4622

This theorem is referenced by:  rankxpsuc 4695

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-r1 4623  df-rank 4624

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