Theorem sucxpdom 4996

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 4618	. . 3 |- (1o ~< A -> (1o e. V /\ A e. V))
2	1	pm3.27d 323	. 2 |- (1o ~< A -> A e. V)
3		breq2 2696	. . . 4 |- (x = A -> (1o ~< x <-> 1o ~< A))
4		suceq 3038	. . . . 5 |- (x = A -> suc x = suc A)
5		xpeq1 3281	. . . . . 6 |- (x = A -> (x X. x) = (A X. x))
6		xpeq2 3282	. . . . . 6 |- (x = A -> (A X. x) = (A X. A))
7	5, 6	eqtrd 1550	. . . . 5 |- (x = A -> (x X. x) = (A X. A))
8	4, 7	breq12d 2704	. . . 4 |- (x = A -> (suc x ~<_ (x X. x) <-> suc A ~<_ (A X. A)))
9	3, 8	imbi12d 629	. . 3 |- (x = A -> ((1o ~< x -> suc x ~<_ (x X. x)) <-> (1o ~< A -> suc A ~<_ (A X. A))))
10		visset 1859	. . . . . . . . . 10 |- x e. V
11		0ex 2785	. . . . . . . . . . 11 |- (/) e. V
12	10, 11	xpsnen 4576	. . . . . . . . . 10 |- (x X. {(/)}) ~~ x
13		sdomen2 4627	. . . . . . . . . 10 |- ((x e. V /\ (x X. {(/)}) ~~ x) -> ({x} ~< (x X. {(/)}) <-> {x} ~< x))
14	10, 12, 13	mp2an 701	. . . . . . . . 9 |- ({x} ~< (x X. {(/)}) <-> {x} ~< x)
15		1on 4274	. . . . . . . . . . 11 |- 1o e. On
16	15	elisseti 1864	. . . . . . . . . 10 |- 1o e. V
17	10	ensn1 4565	. . . . . . . . . 10 |- {x} ~~ 1o
18		sdomen1 4626	. . . . . . . . . 10 |- ((1o e. V /\ {x} ~~ 1o) -> ({x} ~< x <-> 1o ~< x))
19	16, 17, 18	mp2an 701	. . . . . . . . 9 |- ({x} ~< x <-> 1o ~< x)
20	14, 19	bitri 171	. . . . . . . 8 |- ({x} ~< (x X. {(/)}) <-> 1o ~< x)
21		sdomdom 4527	. . . . . . . 8 |- ({x} ~< (x X. {(/)}) -> {x} ~<_ (x X. {(/)}))
22	20, 21	sylbir 199	. . . . . . 7 |- (1o ~< x -> {x} ~<_ (x X. {(/)}))
23		domrefg 4534	. . . . . . . . . 10 |- (x e. V -> x ~<_ x)
24	10, 23	ax-mp 7	. . . . . . . . 9 |- x ~<_ x
25	10, 16	xpsnen 4576	. . . . . . . . . 10 |- (x X. {1o}) ~~ x
26		domen2 4625	. . . . . . . . . 10 |- ((x e. V /\ (x X. {1o}) ~~ x) -> (x ~<_ (x X. {1o}) <-> x ~<_ x))
27	10, 25, 26	mp2an 701	. . . . . . . . 9 |- (x ~<_ (x X. {1o}) <-> x ~<_ x)
28	24, 27	mpbir 188	. . . . . . . 8 |- x ~<_ (x X. {1o})
29		1n0 4278	. . . . . . . . . 10 |- 1o =/= (/)
30		xpsndisj 3555	. . . . . . . . . 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 2826	. . . . . . . . . . 11 |- {1o} e. V
33	10, 32	xpex 3349	. . . . . . . . . 10 |- (x X. {1o}) e. V
34		snex 2826	. . . . . . . . . 10 |- {x} e. V
35		p0ex 2828	. . . . . . . . . . 11 |- {(/)} e. V
36	10, 35	xpex 3349	. . . . . . . . . 10 |- (x X. {(/)}) e. V
37	33, 34, 36	undom 4579	. . . . . . . . 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 700	. . . . . . . 8 |- ((x ~<_ (x X. {1o}) /\ {x} ~<_ (x X. {(/)})) -> (x u. {x}) ~<_ ((x X. {1o}) u. (x X. {(/)})))
39	28, 38	mpan 699	. . . . . . 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 4994	. . . . . . 7 |- ((1o ~< (x X. {1o}) /\ 1o ~< (x X. {(/)})) -> ((x X. {1o}) u. (x X. {(/)})) ~<_ ((x X. {1o}) X. (x X. {(/)})))
42		sdomen2 4627	. . . . . . . 8 |- ((x e. V /\ (x X. {1o}) ~~ x) -> (1o ~< (x X. {1o}) <-> 1o ~< x))
43	10, 25, 42	mp2an 701	. . . . . . 7 |- (1o ~< (x X. {1o}) <-> 1o ~< x)
44		sdomen2 4627	. . . . . . . 8 |- ((x e. V /\ (x X. {(/)}) ~~ x) -> (1o ~< (x X. {(/)}) <-> 1o ~< x))
45	10, 12, 44	mp2an 701	. . . . . . 7 |- (1o ~< (x X. {(/)}) <-> 1o ~< x)
46	41, 43, 45	sylancbr 477	. . . . . 6 |- (1o ~< x -> ((x X. {1o}) u. (x X. {(/)})) ~<_ ((x X. {1o}) X. (x X. {(/)})))
47	40, 46	jca 286	. . . . 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 4556	. . . . 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 4635	. . . . . . 7 |- (((x X. {1o}) ~~ x /\ (x X. {(/)}) ~~ x) -> ((x X. {1o}) X. (x X. {(/)})) ~~ (x X. x))
50	25, 12, 49	mp2an 701	. . . . . 6 |- ((x X. {1o}) X. (x X. {(/)})) ~~ (x X. x)
51		domentr 4562	. . . . . 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 700	. . . . 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 2981	. . . 4 |- suc x = (x u. {x})
55	53, 54	syl5eqbr 2721	. . 3 |- (1o ~< x -> suc x ~<_ (x X. x))
56	9, 55	vtoclg 1893	. 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 144   /\ wa 221   = wceq 992   e. wcel 994   =/= wne 1628  Vcvv 1857   u. cun 2097   i^i cin 2098  (/)c0 2332  {csn 2467   class class class wbr 2692  Oncon0 2975  suc csuc 2977   X. cxp 3249  1oc1o 4264   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-2o 4270  df-er 4401  df-en 4509  df-dom 4510  df-sdom 4511  df-fin 4512  df-card 4962

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