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 × A))

Proof of Theorem sucxpdom

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

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ∩ cin 1486  ∅c0 1707  {csn 1808   class class class wbr 2054  Oncon0 2199  suc csuc 2201   × cxp 2408  1_oc1o 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

metamath.org