Theorem sucxpdom 3652

Description: Cross product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals, with a proof using AC).

Assertion

Ref	Expression
sucxpdom	|- (1o ~< A -> suc A ~<_ (A X. A))

Proof of Theorem sucxpdom

Step	Hyp	Ref	Expression
1		sdomex 3315	. . 3 |- (1o ~< A -> (1o e. V /\ A e. V))
2	1	pm3.27d 262	. 2 |- (1o ~< A -> A e. V)
3		breq2 2066	. . . 4 |- (x = A -> (1o ~< x <-> 1o ~< A))
4		suceq 2288	. . . . 5 |- (x = A -> suc x = suc A)
5		xpeq1 2440	. . . . . 6 |- (x = A -> (x X. x) = (A X. x))
6		xpeq2 2441	. . . . . 6 |- (x = A -> (A X. x) = (A X. A))
7	5, 6	eqtrd 1128	. . . . 5 |- (x = A -> (x X. x) = (A X. A))
8	4, 7	breq12d 2073	. . . 4 |- (x = A -> (suc x ~<_ (x X. x) <-> suc A ~<_ (A X. A)))
9	3, 8	imbi12d 474	. . 3 |- (x = A -> ((1o ~< x -> suc x ~<_ (x X. x)) <-> (1o ~< A -> suc A ~<_ (A X. A))))
10		visset 1350	. . . . . . . . . 10 |- x e. V
11		1o 3109	. . . . . . . . . . . 12 |- 1o e. On
12	11	elisseti 1355	. . . . . . . . . . 11 |- 1o e. V
13	10, 12	xpsnen 3339	. . . . . . . . . 10 |- (x X. {1o}) ~~ x
14		sdomen2 3380	. . . . . . . . . 10 |- ((x e. V /\ (x X. {1o}) ~~ x) -> ({x} ~< (x X. {1o}) <-> {x} ~< x))
15	10, 13, 14	mp2an 520	. . . . . . . . 9 |- ({x} ~< (x X. {1o}) <-> {x} ~< x)
16	10	ensn1 3329	. . . . . . . . . 10 |- {x} ~~ 1o
17		sdomen1 3379	. . . . . . . . . 10 |- ((1o e. V /\ {x} ~~ 1o) -> ({x} ~< x <-> 1o ~< x))
18	12, 16, 17	mp2an 520	. . . . . . . . 9 |- ({x} ~< x <-> 1o ~< x)
19	15, 18	bitr 151	. . . . . . . 8 |- ({x} ~< (x X. {1o}) <-> 1o ~< x)
20		sdomdom 3290	. . . . . . . 8 |- ({x} ~< (x X. {1o}) -> {x} ~<_ (x X. {1o}))
21	19, 20	sylbir 176	. . . . . . 7 |- (1o ~< x -> {x} ~<_ (x X. {1o}))
22		domrefg 3297	. . . . . . . . . 10 |- (x e. V -> x ~<_ x)
23	10, 22	ax-mp 6	. . . . . . . . 9 |- x ~<_ x
24		0ex 1745	. . . . . . . . . . 11 |- (/) e. V
25	10, 24	xpsnen 3339	. . . . . . . . . 10 |- (x X. {(/)}) ~~ x
26		domen2 3378	. . . . . . . . . 10 |- ((x e. V /\ (x X. {(/)}) ~~ x) -> (x ~<_ (x X. {(/)}) <-> x ~<_ x))
27	10, 25, 26	mp2an 520	. . . . . . . . 9 |- (x ~<_ (x X. {(/)}) <-> x ~<_ x)
28	23, 27	mpbir 165	. . . . . . . 8 |- x ~<_ (x X. {(/)})
29		0ne1oOLD 3113	. . . . . . . . . 10 |- -. (/) = 1o
30		xpsndisj 2655	. . . . . . . . . 10 |- (-. (/) = 1o -> ((x X. {(/)}) i^i (x X. {1o})) = (/))
31	29, 30	ax-mp 6	. . . . . . . . 9 |- ((x X. {(/)}) i^i (x X. {1o})) = (/)
32		p0ex 1885	. . . . . . . . . . 11 |- {(/)} e. V
33	10, 32	xpex 2488	. . . . . . . . . 10 |- (x X. {(/)}) e. V
34		snex 1859	. . . . . . . . . 10 |- {x} e. V
35		snex 1859	. . . . . . . . . . 11 |- {1o} e. V
36	10, 35	xpex 2488	. . . . . . . . . 10 |- (x X. {1o}) e. V
37	33, 34, 36	undom 3342	. . . . . . . . 9 |- (((x ~<_ (x X. {(/)}) /\ {x} ~<_ (x X. {1o})) /\ ((x X. {(/)}) i^i (x X. {1o})) = (/)) -> (x u. {x}) ~<_ ((x X. {(/)}) u. (x X. {1o})))
38	31, 37	mpan2 519	. . . . . . . 8 |- ((x ~<_ (x X. {(/)}) /\ {x} ~<_ (x X. {1o})) -> (x u. {x}) ~<_ ((x X. {(/)}) u. (x X. {1o})))
39	28, 38	mpan 518	. . . . . . 7 |- ({x} ~<_ (x X. {1o}) -> (x u. {x}) ~<_ ((x X. {(/)}) u. (x X. {1o})))
40	21, 39	syl 12	. . . . . 6 |- (1o ~< x -> (x u. {x}) ~<_ ((x X. {(/)}) u. (x X. {1o})))
41		unxpdom 3650	. . . . . . 7 |- ((1o ~< (x X. {(/)}) /\ 1o ~< (x X. {1o})) -> ((x X. {(/)}) u. (x X. {1o})) ~<_ ((x X. {(/)}) X. (x X. {1o})))
42		sdomen2 3380	. . . . . . . 8 |- ((x e. V /\ (x X. {(/)}) ~~ x) -> (1o ~< (x X. {(/)}) <-> 1o ~< x))
43	10, 25, 42	mp2an 520	. . . . . . 7 |- (1o ~< (x X. {(/)}) <-> 1o ~< x)
44		sdomen2 3380	. . . . . . . 8 |- ((x e. V /\ (x X. {1o}) ~~ x) -> (1o ~< (x X. {1o}) <-> 1o ~< x))
45	10, 13, 44	mp2an 520	. . . . . . 7 |- (1o ~< (x X. {1o}) <-> 1o ~< x)
46	41, 43, 45	sylancbr 363	. . . . . 6 |- (1o ~< x -> ((x X. {(/)}) u. (x X. {1o})) ~<_ ((x X. {(/)}) X. (x X. {1o})))
47	40, 46	jca 236	. . . . 5 |- (1o ~< x -> ((x u. {x}) ~<_ ((x X. {(/)}) u. (x X. {1o})) /\ ((x X. {(/)}) u. (x X. {1o})) ~<_ ((x X. {(/)}) X. (x X. {1o}))))
48		domtr 3320	. . . . 5 |- (((x u. {x}) ~<_ ((x X. {(/)}) u. (x X. {1o})) /\ ((x X. {(/)}) u. (x X. {1o})) ~<_ ((x X. {(/)}) X. (x X. {1o}))) -> (x u. {x}) ~<_ ((x X. {(/)}) X. (x X. {1o})))
49	33, 10, 36, 10	xpen 3383	. . . . . . 7 |- (((x X. {(/)}) ~~ x /\ (x X. {1o}) ~~ x) -> ((x X. {(/)}) X. (x X. {1o})) ~~ (x X. x))
50	25, 13, 49	mp2an 520	. . . . . 6 |- ((x X. {(/)}) X. (x X. {1o})) ~~ (x X. x)
51		domentr 3326	. . . . . 6 |- (((x u. {x}) ~<_ ((x X. {(/)}) X. (x X. {1o})) /\ ((x X. {(/)}) X. (x X. {1o})) ~~ (x X. x)) -> (x u. {x}) ~<_ (x X. x))
52	50, 51	mpan2 519	. . . . 5 |- ((x u. {x}) ~<_ ((x X. {(/)}) X. (x X. {1o})) -> (x u. {x}) ~<_ (x X. x))
53	47, 48, 52	3syl 21	. . . 4 |- (1o ~< x -> (x u. {x}) ~<_ (x X. x))
54		df-suc 2205	. . . 4 |- suc x = (x u. {x})
55	53, 54	syl5eqbr 2089	. . 3 |- (1o ~< x -> suc x ~<_ (x X. x))
56	9, 55	vtoclg 1383	. 2 |- (A e. V -> (1o ~< A -> suc A ~<_ (A X. A)))
57	2, 56	mpcom 49	1 |- (1o ~< A -> suc A ~<_ (A X. A))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   u. cun 1485   i^i cin 1486  (/)c0 1707  {csn 1808   class class class wbr 2054  Oncon0 2199  suc csuc 2201   X. cxp 2408  1oc1o 3099   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-2o 3105  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623

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