Theorem sucxpdom 5998

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 5536	. . 3 |- (1o ~< A -> (1o e. _V /\ A e. _V))
2	1	simprd 352	. 2 |- (1o ~< A -> A e. _V)
3		breq2 3342	. . . 4 |- (x = A -> (1o ~< x <-> 1o ~< A))
4		suceq 3729	. . . . 5 |- (x = A -> suc x = suc A)
5		xpeq1 4016	. . . . . 6 |- (x = A -> (x X. x) = (A X. x))
6		xpeq2 4017	. . . . . 6 |- (x = A -> (A X. x) = (A X. A))
7	5, 6	eqtrd 1925	. . . . 5 |- (x = A -> (x X. x) = (A X. A))
8	4, 7	breq12d 3351	. . . 4 |- (x = A -> (suc x ~<_ (x X. x) <-> suc A ~<_ (A X. A)))
9	3, 8	imbi12d 688	. . 3 |- (x = A -> ((1o ~< x -> suc x ~<_ (x X. x)) <-> (1o ~< A -> suc A ~<_ (A X. A))))
10		1n0 5187	. . . . . . . . 9 |- 1o =/= (/)
11		xpsndisj 4339	. . . . . . . . 9 |- (1o =/= (/) -> ((x X. {1o}) i^i (x X. {(/)})) = (/))
12	10, 11	ax-mp 7	. . . . . . . 8 |- ((x X. {1o}) i^i (x X. {(/)})) = (/)
13		visset 2295	. . . . . . . . . 10 |- x e. _V
14		snex 3492	. . . . . . . . . 10 |- {1o} e. _V
15	13, 14	xpex 4096	. . . . . . . . 9 |- (x X. {1o}) e. _V
16		snex 3492	. . . . . . . . 9 |- {x} e. _V
17		p0ex 3495	. . . . . . . . . 10 |- {(/)} e. _V
18	13, 17	xpex 4096	. . . . . . . . 9 |- (x X. {(/)}) e. _V
19	15, 16, 18	undom 5497	. . . . . . . 8 |- (((x ~<_ (x X. {1o}) /\ {x} ~<_ (x X. {(/)})) /\ ((x X. {1o}) i^i (x X. {(/)})) = (/)) -> (x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})))
20	12, 19	mpan2 760	. . . . . . 7 |- ((x ~<_ (x X. {1o}) /\ {x} ~<_ (x X. {(/)})) -> (x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})))
21		domrefg 5452	. . . . . . . . 9 |- (x e. _V -> x ~<_ x)
22	13, 21	ax-mp 7	. . . . . . . 8 |- x ~<_ x
23		1on 5182	. . . . . . . . . . 11 |- 1o e. On
24	23	elisseti 2301	. . . . . . . . . 10 |- 1o e. _V
25	13, 24	xpsnen 5494	. . . . . . . . 9 |- (x X. {1o}) ~~ x
26		domen2 5543	. . . . . . . . 9 |- ((x e. _V /\ (x X. {1o}) ~~ x) -> (x ~<_ (x X. {1o}) <-> x ~<_ x))
27	13, 25, 26	mp2an 761	. . . . . . . 8 |- (x ~<_ (x X. {1o}) <-> x ~<_ x)
28	22, 27	mpbir 207	. . . . . . 7 |- x ~<_ (x X. {1o})
29		0ex 3446	. . . . . . . . . . 11 |- (/) e. _V
30	13, 29	xpsnen 5494	. . . . . . . . . 10 |- (x X. {(/)}) ~~ x
31		sdomen2 5545	. . . . . . . . . 10 |- ((x e. _V /\ (x X. {(/)}) ~~ x) -> ({x} ~< (x X. {(/)}) <-> {x} ~< x))
32	13, 30, 31	mp2an 761	. . . . . . . . 9 |- ({x} ~< (x X. {(/)}) <-> {x} ~< x)
33	13	ensn1 5483	. . . . . . . . . 10 |- {x} ~~ 1o
34		sdomen1 5544	. . . . . . . . . 10 |- ((1o e. _V /\ {x} ~~ 1o) -> ({x} ~< x <-> 1o ~< x))
35	24, 33, 34	mp2an 761	. . . . . . . . 9 |- ({x} ~< x <-> 1o ~< x)
36	32, 35	bitri 190	. . . . . . . 8 |- ({x} ~< (x X. {(/)}) <-> 1o ~< x)
37		sdomdom 5445	. . . . . . . 8 |- ({x} ~< (x X. {(/)}) -> {x} ~<_ (x X. {(/)}))
38	36, 37	sylbir 218	. . . . . . 7 |- (1o ~< x -> {x} ~<_ (x X. {(/)}))
39	20, 28, 38	sylancr 526	. . . . . 6 |- (1o ~< x -> (x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})))
40		unxpdom 5996	. . . . . . 7 |- ((1o ~< (x X. {1o}) /\ 1o ~< (x X. {(/)})) -> ((x X. {1o}) u. (x X. {(/)})) ~<_ ((x X. {1o}) X. (x X. {(/)})))
41		sdomen2 5545	. . . . . . . 8 |- ((x e. _V /\ (x X. {1o}) ~~ x) -> (1o ~< (x X. {1o}) <-> 1o ~< x))
42	13, 25, 41	mp2an 761	. . . . . . 7 |- (1o ~< (x X. {1o}) <-> 1o ~< x)
43		sdomen2 5545	. . . . . . . 8 |- ((x e. _V /\ (x X. {(/)}) ~~ x) -> (1o ~< (x X. {(/)}) <-> 1o ~< x))
44	13, 30, 43	mp2an 761	. . . . . . 7 |- (1o ~< (x X. {(/)}) <-> 1o ~< x)
45	40, 42, 44	sylancbr 530	. . . . . 6 |- (1o ~< x -> ((x X. {1o}) u. (x X. {(/)})) ~<_ ((x X. {1o}) X. (x X. {(/)})))
46	39, 45	jca 310	. . . . 5 |- (1o ~< x -> ((x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})) /\ ((x X. {1o}) u. (x X. {(/)})) ~<_ ((x X. {1o}) X. (x X. {(/)}))))
47		domtr 5474	. . . . 5 |- (((x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})) /\ ((x X. {1o}) u. (x X. {(/)})) ~<_ ((x X. {1o}) X. (x X. {(/)}))) -> (x u. {x}) ~<_ ((x X. {1o}) X. (x X. {(/)})))
48	15, 13, 18, 13	xpen 5582	. . . . . . 7 |- (((x X. {1o}) ~~ x /\ (x X. {(/)}) ~~ x) -> ((x X. {1o}) X. (x X. {(/)})) ~~ (x X. x))
49	25, 30, 48	mp2an 761	. . . . . 6 |- ((x X. {1o}) X. (x X. {(/)})) ~~ (x X. x)
50		domentr 5480	. . . . . 6 |- (((x u. {x}) ~<_ ((x X. {1o}) X. (x X. {(/)})) /\ ((x X. {1o}) X. (x X. {(/)})) ~~ (x X. x)) -> (x u. {x}) ~<_ (x X. x))
51	49, 50	mpan2 760	. . . . 5 |- ((x u. {x}) ~<_ ((x X. {1o}) X. (x X. {(/)})) -> (x u. {x}) ~<_ (x X. x))
52	46, 47, 51	3syl 24	. . . 4 |- (1o ~< x -> (x u. {x}) ~<_ (x X. x))
53		df-suc 3663	. . . 4 |- suc x = (x u. {x})
54	52, 53	syl5eqbr 3370	. . 3 |- (1o ~< x -> suc x ~<_ (x X. x))
55	9, 54	vtoclg 2346	. 2 |- (A e. _V -> (1o ~< A -> suc A ~<_ (A X. A)))
56	2, 55	mpcom 60	1 |- (1o ~< A -> suc A ~<_ (A X. A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044   class class class wbr 3338  Oncon0 3657  suc csuc 3659   X. cxp 3984  1oc1o 5172   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425

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  ax-inf2 5731  ax-ac 5906

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-reu 2111  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-int 3215  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-om 3950  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-2o 5178  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862

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