Theorem sucxpdom 9228

Description: Cartesian 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). (Contributed by NM, 3-Sep-2004.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
sucxpdom	⊢ (1o ≺ 𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

Proof of Theorem sucxpdom

Step	Hyp	Ref	Expression
1		df-suc 6370	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		relsdom 8956	. . . . . . . . 9 ⊢ Rel ≺
3	2	brrelex2i 5720	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
4		1on 8472	. . . . . . . 8 ⊢ 1o ∈ On
5		xpsneng 9057	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
6	3, 4, 5	sylancl 598	. . . . . . 7 ⊢ (1o ≺ 𝐴 → (𝐴 × {1o}) ≈ 𝐴)
7	6	ensymd 9008	. . . . . 6 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {1o}))
8		endom 8982	. . . . . 6 ⊢ (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
9	7, 8	syl 18	. . . . 5 ⊢ (1o ≺ 𝐴 → 𝐴 ≼ (𝐴 × {1o}))
10		ensn1g 9025	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o)
11	3, 10	syl 18	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → {𝐴} ≈ 1o)
12		ensdomtr 9108	. . . . . . . 8 ⊢ (({𝐴} ≈ 1o ∧ 1o ≺ 𝐴) → {𝐴} ≺ 𝐴)
13	11, 12	mpancom 701	. . . . . . 7 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ 𝐴)
14		0ex 5272	. . . . . . . . 9 ⊢ ∅ ∈ V
15		xpsneng 9057	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
16	3, 14, 15	sylancl 598	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
17	16	ensymd 9008	. . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {∅}))
18		sdomentr 9106	. . . . . . 7 ⊢ (({𝐴} ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
19	13, 17, 18	syl2anc 596	. . . . . 6 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20		sdomdom 8983	. . . . . 6 ⊢ ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
21	19, 20	syl 18	. . . . 5 ⊢ (1o ≺ 𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22		1n0 8478	. . . . . 6 ⊢ 1o ≠ ∅
23		xpsndisj 6162	. . . . . 6 ⊢ (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
24	22, 23	mp1i 14	. . . . 5 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25		undom 9060	. . . . 5 ⊢ (((𝐴 ≼ (𝐴 × {1o}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
26	9, 21, 24, 25	syl21anc 851	. . . 4 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
27		sdomentr 9106	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
28	7, 27	mpdan 700	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {1o}))
29		sdomentr 9106	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
30	17, 29	mpdan 700	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {∅}))
31		unxpdom 9226	. . . . 5 ⊢ ((1o ≺ (𝐴 × {1o}) ∧ 1o ≺ (𝐴 × {∅})) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
32	28, 30, 31	syl2anc 596	. . . 4 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
33		domtr 9010	. . . 4 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
34	26, 32, 33	syl2anc 596	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
35		xpen 9135	. . . 4 ⊢ (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
36	6, 16, 35	syl2anc 596	. . 3 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37		domentr 9016	. . 3 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})) ∧ ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
38	34, 36, 37	syl2anc 596	. 2 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
39	1, 38	eqbrtrid 5148	1 ⊢ (1o ≺ 𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457   ∪ cun 3904   ∩ cin 3905  ∅c0 4286  {csn 4591   class class class wbr 5111   × cxp 5661  Oncon0 6364  suc csuc 6366  1_oc1o 8452   ≈ cen 8946   ≼ cdom 8947   ≺ csdm 8948

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1st 7992  df-2nd 7993  df-1o 8459  df-2o 8460  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952

This theorem is used by: (None)