Theorem sucxpdom 4818

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 4453	. . 3 |- (1o ~< A -> (1o e. V /\ A e. V))
2	1	pm3.27d 325	. 2 |- (1o ~< A -> A e. V)
3		breq2 2613	. . . 4 |- (x = A -> (1o ~< x <-> 1o ~< A))
4		suceq 3024	. . . . 5 |- (x = A -> suc x = suc A)
5		xpeq1 3190	. . . . . 6 |- (x = A -> (x X. x) = (A X. x))
6		xpeq2 3191	. . . . . 6 |- (x = A -> (A X. x) = (A X. A))
7	5, 6	eqtrd 1499	. . . . 5 |- (x = A -> (x X. x) = (A X. A))
8	4, 7	breq12d 2621	. . . 4 |- (x = A -> (suc x ~<_ (x X. x) <-> suc A ~<_ (A X. A)))
9	3, 8	imbi12d 624	. . 3 |- (x = A -> ((1o ~< x -> suc x ~<_ (x X. x)) <-> (1o ~< A -> suc A ~<_ (A X. A))))
10		visset 1804	. . . . . . . . . 10 |- x e. V
11		0ex 2701	. . . . . . . . . . 11 |- (/) e. V
12	10, 11	xpsnen 4415	. . . . . . . . . 10 |- (x X. {(/)}) ~~ x
13		sdomen2 4462	. . . . . . . . . 10 |- ((x e. V /\ (x X. {(/)}) ~~ x) -> ({x} ~< (x X. {(/)}) <-> {x} ~< x))
14	10, 12, 13	mp2an 695	. . . . . . . . 9 |- ({x} ~< (x X. {(/)}) <-> {x} ~< x)
15		1on 4122	. . . . . . . . . . 11 |- 1o e. On
16	15	elisseti 1809	. . . . . . . . . 10 |- 1o e. V
17	10	ensn1 4405	. . . . . . . . . 10 |- {x} ~~ 1o
18		sdomen1 4461	. . . . . . . . . 10 |- ((1o e. V /\ {x} ~~ 1o) -> ({x} ~< x <-> 1o ~< x))
19	16, 17, 18	mp2an 695	. . . . . . . . 9 |- ({x} ~< x <-> 1o ~< x)
20	14, 19	bitr 173	. . . . . . . 8 |- ({x} ~< (x X. {(/)}) <-> 1o ~< x)
21		sdomdom 4367	. . . . . . . 8 |- ({x} ~< (x X. {(/)}) -> {x} ~<_ (x X. {(/)}))
22	20, 21	sylbir 201	. . . . . . 7 |- (1o ~< x -> {x} ~<_ (x X. {(/)}))
23		domrefg 4374	. . . . . . . . . 10 |- (x e. V -> x ~<_ x)
24	10, 23	ax-mp 7	. . . . . . . . 9 |- x ~<_ x
25	10, 16	xpsnen 4415	. . . . . . . . . 10 |- (x X. {1o}) ~~ x
26		domen2 4460	. . . . . . . . . 10 |- ((x e. V /\ (x X. {1o}) ~~ x) -> (x ~<_ (x X. {1o}) <-> x ~<_ x))
27	10, 25, 26	mp2an 695	. . . . . . . . 9 |- (x ~<_ (x X. {1o}) <-> x ~<_ x)
28	24, 27	mpbir 190	. . . . . . . 8 |- x ~<_ (x X. {1o})
29		1ne0 4126	. . . . . . . . . 10 |- 1o =/= (/)
30		xpsndisj 3456	. . . . . . . . . 10 |- (1o =/= (/) -> ((x X. {1o}) i^i (x X. {(/)})) = (/))
31	29, 30	ax-mp 7	. . . . . . . . 9 |- ((x X. {1o}) i^i (x X. {(/)})) = (/)
32		snex 2740	. . . . . . . . . . 11 |- {1o} e. V
33	10, 32	xpex 3250	. . . . . . . . . 10 |- (x X. {1o}) e. V
34		snex 2740	. . . . . . . . . 10 |- {x} e. V
35		p0ex 2760	. . . . . . . . . . 11 |- {(/)} e. V
36	10, 35	xpex 3250	. . . . . . . . . 10 |- (x X. {(/)}) e. V
37	33, 34, 36	undom 4418	. . . . . . . . 9 |- (((x ~<_ (x X. {1o}) /\ {x} ~<_ (x X. {(/)})) /\ ((x X. {1o}) i^i (x X. {(/)})) = (/)) -> (x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})))
38	31, 37	mpan2 694	. . . . . . . 8 |- ((x ~<_ (x X. {1o}) /\ {x} ~<_ (x X. {(/)})) -> (x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})))
39	28, 38	mpan 693	. . . . . . 7 |- ({x} ~<_ (x X. {(/)}) -> (x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})))
40	22, 39	syl 10	. . . . . 6 |- (1o ~< x -> (x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})))
41		unxpdom 4816	. . . . . . 7 |- ((1o ~< (x X. {1o}) /\ 1o ~< (x X. {(/)})) -> ((x X. {1o}) u. (x X. {(/)})) ~<_ ((x X. {1o}) X. (x X. {(/)})))
42		sdomen2 4462	. . . . . . . 8 |- ((x e. V /\ (x X. {1o}) ~~ x) -> (1o ~< (x X. {1o}) <-> 1o ~< x))
43	10, 25, 42	mp2an 695	. . . . . . 7 |- (1o ~< (x X. {1o}) <-> 1o ~< x)
44		sdomen2 4462	. . . . . . . 8 |- ((x e. V /\ (x X. {(/)}) ~~ x) -> (1o ~< (x X. {(/)}) <-> 1o ~< x))
45	10, 12, 44	mp2an 695	. . . . . . 7 |- (1o ~< (x X. {(/)}) <-> 1o ~< x)
46	41, 43, 45	sylancbr 474	. . . . . 6 |- (1o ~< x -> ((x X. {1o}) u. (x X. {(/)})) ~<_ ((x X. {1o}) X. (x X. {(/)})))
47	40, 46	jca 288	. . . . 5 |- (1o ~< x -> ((x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})) /\ ((x X. {1o}) u. (x X. {(/)})) ~<_ ((x X. {1o}) X. (x X. {(/)}))))
48		domtr 4396	. . . . 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. {(/)})))
49	33, 10, 36, 10	xpen 4468	. . . . . . 7 |- (((x X. {1o}) ~~ x /\ (x X. {(/)}) ~~ x) -> ((x X. {1o}) X. (x X. {(/)})) ~~ (x X. x))
50	25, 12, 49	mp2an 695	. . . . . 6 |- ((x X. {1o}) X. (x X. {(/)})) ~~ (x X. x)
51		domentr 4402	. . . . . 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))
52	50, 51	mpan2 694	. . . . 5 |- ((x u. {x}) ~<_ ((x X. {1o}) X. (x X. {(/)})) -> (x u. {x}) ~<_ (x X. x))
53	47, 48, 52	3syl 20	. . . 4 |- (1o ~< x -> (x u. {x}) ~<_ (x X. x))
54		df-suc 2944	. . . 4 |- suc x = (x u. {x})
55	53, 54	syl5eqbr 2638	. . 3 |- (1o ~< x -> suc x ~<_ (x X. x))
56	9, 55	vtoclg 1838	. 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:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  Vcvv 1802   u. cun 2035   i^i cin 2036  (/)c0 2270  {csn 2399   class class class wbr 2609  Oncon0 2938  suc csuc 2940   X. cxp 3158  1oc1o 4112   ~~ cen 4348   ~<_ cdom 4349   ~< csdm 4350

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-2o 4118  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788

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