Theorem sucxpdom 9261

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 8952	. . . . . . . . 9 ⊢ Rel ≺
3	2	brrelex2i 5733	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
4		1on 8484	. . . . . . . 8 ⊢ 1o ∈ On
5		xpsneng 9062	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
6	3, 4, 5	sylancl 585	. . . . . . 7 ⊢ (1o ≺ 𝐴 → (𝐴 × {1o}) ≈ 𝐴)
7	6	ensymd 9007	. . . . . 6 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {1o}))
8		endom 8981	. . . . . 6 ⊢ (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
9	7, 8	syl 17	. . . . 5 ⊢ (1o ≺ 𝐴 → 𝐴 ≼ (𝐴 × {1o}))
10		ensn1g 9025	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o)
11	3, 10	syl 17	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → {𝐴} ≈ 1o)
12		ensdomtr 9119	. . . . . . . 8 ⊢ (({𝐴} ≈ 1o ∧ 1o ≺ 𝐴) → {𝐴} ≺ 𝐴)
13	11, 12	mpancom 685	. . . . . . 7 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ 𝐴)
14		0ex 5307	. . . . . . . . 9 ⊢ ∅ ∈ V
15		xpsneng 9062	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
16	3, 14, 15	sylancl 585	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
17	16	ensymd 9007	. . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {∅}))
18		sdomentr 9117	. . . . . . 7 ⊢ (({𝐴} ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
19	13, 17, 18	syl2anc 583	. . . . . 6 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20		sdomdom 8982	. . . . . 6 ⊢ ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
21	19, 20	syl 17	. . . . 5 ⊢ (1o ≺ 𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22		1n0 8494	. . . . . 6 ⊢ 1o ≠ ∅
23		xpsndisj 6162	. . . . . 6 ⊢ (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
24	22, 23	mp1i 13	. . . . 5 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25		undom 9065	. . . . 5 ⊢ (((𝐴 ≼ (𝐴 × {1o}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
26	9, 21, 24, 25	syl21anc 835	. . . 4 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
27		sdomentr 9117	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
28	7, 27	mpdan 684	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {1o}))
29		sdomentr 9117	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
30	17, 29	mpdan 684	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {∅}))
31		unxpdom 9259	. . . . 5 ⊢ ((1o ≺ (𝐴 × {1o}) ∧ 1o ≺ (𝐴 × {∅})) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
32	28, 30, 31	syl2anc 583	. . . 4 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
33		domtr 9009	. . . 4 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
34	26, 32, 33	syl2anc 583	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
35		xpen 9146	. . . 4 ⊢ (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
36	6, 16, 35	syl2anc 583	. . 3 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37		domentr 9015	. . 3 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})) ∧ ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
38	34, 36, 37	syl2anc 583	. 2 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
39	1, 38	eqbrtrid 5183	1 ⊢ (1o ≺ 𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  Vcvv 3473   ∪ cun 3946   ∩ cin 3947  ∅c0 4322  {csn 4628   class class class wbr 5148   × cxp 5674  Oncon0 6364  suc csuc 6366  1_oc1o 8465   ≈ cen 8942   ≼ cdom 8943   ≺ csdm 8944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 7979  df-2nd 7980  df-1o 8472  df-2o 8473  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948

This theorem is referenced by: (None)