Theorem sucxpdom 8169

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 5729	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		relsdom 7962	. . . . . . . . 9 ⊢ Rel ≺
3	2	brrelex2i 5159	. . . . . . . 8 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ∈ V)
4		1on 7567	. . . . . . . 8 ⊢ 1𝑜 ∈ On
5		xpsneng 8045	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
6	3, 4, 5	sylancl 694	. . . . . . 7 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 × {1𝑜}) ≈ 𝐴)
7	6	ensymd 8007	. . . . . 6 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ≈ (𝐴 × {1𝑜}))
8		endom 7982	. . . . . 6 ⊢ (𝐴 ≈ (𝐴 × {1𝑜}) → 𝐴 ≼ (𝐴 × {1𝑜}))
9	7, 8	syl 17	. . . . 5 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ≼ (𝐴 × {1𝑜}))
10		ensn1g 8021	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
11	3, 10	syl 17	. . . . . . . 8 ⊢ (1𝑜 ≺ 𝐴 → {𝐴} ≈ 1𝑜)
12		ensdomtr 8096	. . . . . . . 8 ⊢ (({𝐴} ≈ 1𝑜 ∧ 1𝑜 ≺ 𝐴) → {𝐴} ≺ 𝐴)
13	11, 12	mpancom 703	. . . . . . 7 ⊢ (1𝑜 ≺ 𝐴 → {𝐴} ≺ 𝐴)
14		0ex 4790	. . . . . . . . 9 ⊢ ∅ ∈ V
15		xpsneng 8045	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
16	3, 14, 15	sylancl 694	. . . . . . . 8 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
17	16	ensymd 8007	. . . . . . 7 ⊢ (1𝑜 ≺ 𝐴 → 𝐴 ≈ (𝐴 × {∅}))
18		sdomentr 8094	. . . . . . 7 ⊢ (({𝐴} ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
19	13, 17, 18	syl2anc 693	. . . . . 6 ⊢ (1𝑜 ≺ 𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20		sdomdom 7983	. . . . . 6 ⊢ ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
21	19, 20	syl 17	. . . . 5 ⊢ (1𝑜 ≺ 𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22		1n0 7575	. . . . . 6 ⊢ 1𝑜 ≠ ∅
23		xpsndisj 5557	. . . . . 6 ⊢ (1𝑜 ≠ ∅ → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
24	22, 23	mp1i 13	. . . . 5 ⊢ (1𝑜 ≺ 𝐴 → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
25		undom 8048	. . . . 5 ⊢ (((𝐴 ≼ (𝐴 × {1𝑜}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
26	9, 21, 24, 25	syl21anc 1325	. . . 4 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
27		sdomentr 8094	. . . . . 6 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {1𝑜})) → 1𝑜 ≺ (𝐴 × {1𝑜}))
28	7, 27	mpdan 702	. . . . 5 ⊢ (1𝑜 ≺ 𝐴 → 1𝑜 ≺ (𝐴 × {1𝑜}))
29		sdomentr 8094	. . . . . 6 ⊢ ((1𝑜 ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → 1𝑜 ≺ (𝐴 × {∅}))
30	17, 29	mpdan 702	. . . . 5 ⊢ (1𝑜 ≺ 𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
31		unxpdom 8167	. . . . 5 ⊢ ((1𝑜 ≺ (𝐴 × {1𝑜}) ∧ 1𝑜 ≺ (𝐴 × {∅})) → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
32	28, 30, 31	syl2anc 693	. . . 4 ⊢ (1𝑜 ≺ 𝐴 → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
33		domtr 8009	. . . 4 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
34	26, 32, 33	syl2anc 693	. . 3 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
35		xpen 8123	. . . 4 ⊢ (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
36	6, 16, 35	syl2anc 693	. . 3 ⊢ (1𝑜 ≺ 𝐴 → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37		domentr 8015	. . 3 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
38	34, 36, 37	syl2anc 693	. 2 ⊢ (1𝑜 ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
39	1, 38	syl5eqbr 4688	1 ⊢ (1𝑜 ≺ 𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  Oncon0 5723  suc csuc 5725  1_𝑜c1o 7553   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by: (None)