Theorem sucxpdom 9318

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 6401	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		relsdom 9010	. . . . . . . . 9 ⊢ Rel ≺
3	2	brrelex2i 5757	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
4		1on 8534	. . . . . . . 8 ⊢ 1o ∈ On
5		xpsneng 9122	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
6	3, 4, 5	sylancl 585	. . . . . . 7 ⊢ (1o ≺ 𝐴 → (𝐴 × {1o}) ≈ 𝐴)
7	6	ensymd 9065	. . . . . 6 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {1o}))
8		endom 9039	. . . . . 6 ⊢ (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
9	7, 8	syl 17	. . . . 5 ⊢ (1o ≺ 𝐴 → 𝐴 ≼ (𝐴 × {1o}))
10		ensn1g 9084	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o)
11	3, 10	syl 17	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → {𝐴} ≈ 1o)
12		ensdomtr 9179	. . . . . . . 8 ⊢ (({𝐴} ≈ 1o ∧ 1o ≺ 𝐴) → {𝐴} ≺ 𝐴)
13	11, 12	mpancom 687	. . . . . . 7 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ 𝐴)
14		0ex 5325	. . . . . . . . 9 ⊢ ∅ ∈ V
15		xpsneng 9122	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
16	3, 14, 15	sylancl 585	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
17	16	ensymd 9065	. . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {∅}))
18		sdomentr 9177	. . . . . . 7 ⊢ (({𝐴} ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
19	13, 17, 18	syl2anc 583	. . . . . 6 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20		sdomdom 9040	. . . . . 6 ⊢ ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
21	19, 20	syl 17	. . . . 5 ⊢ (1o ≺ 𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22		1n0 8544	. . . . . 6 ⊢ 1o ≠ ∅
23		xpsndisj 6194	. . . . . 6 ⊢ (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
24	22, 23	mp1i 13	. . . . 5 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25		undom 9125	. . . . 5 ⊢ (((𝐴 ≼ (𝐴 × {1o}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
26	9, 21, 24, 25	syl21anc 837	. . . 4 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
27		sdomentr 9177	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
28	7, 27	mpdan 686	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {1o}))
29		sdomentr 9177	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
30	17, 29	mpdan 686	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {∅}))
31		unxpdom 9316	. . . . 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 9067	. . . 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 9206	. . . 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 9073	. . 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 5201	1 ⊢ (1o ≺ 𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488   ∪ cun 3974   ∩ cin 3975  ∅c0 4352  {csn 4648   class class class wbr 5166   × cxp 5698  Oncon0 6395  suc csuc 6397  1_oc1o 8515   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1st 8030  df-2nd 8031  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006

This theorem is referenced by: (None)