Theorem sucxpdom 9263

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 6358	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		relsdom 8966	. . . . . . . . 9 ⊢ Rel ≺
3	2	brrelex2i 5711	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
4		1on 8492	. . . . . . . 8 ⊢ 1o ∈ On
5		xpsneng 9070	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
6	3, 4, 5	sylancl 586	. . . . . . 7 ⊢ (1o ≺ 𝐴 → (𝐴 × {1o}) ≈ 𝐴)
7	6	ensymd 9019	. . . . . 6 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {1o}))
8		endom 8993	. . . . . 6 ⊢ (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
9	7, 8	syl 17	. . . . 5 ⊢ (1o ≺ 𝐴 → 𝐴 ≼ (𝐴 × {1o}))
10		ensn1g 9036	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o)
11	3, 10	syl 17	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → {𝐴} ≈ 1o)
12		ensdomtr 9127	. . . . . . . 8 ⊢ (({𝐴} ≈ 1o ∧ 1o ≺ 𝐴) → {𝐴} ≺ 𝐴)
13	11, 12	mpancom 688	. . . . . . 7 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ 𝐴)
14		0ex 5277	. . . . . . . . 9 ⊢ ∅ ∈ V
15		xpsneng 9070	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
16	3, 14, 15	sylancl 586	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
17	16	ensymd 9019	. . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {∅}))
18		sdomentr 9125	. . . . . . 7 ⊢ (({𝐴} ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
19	13, 17, 18	syl2anc 584	. . . . . 6 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20		sdomdom 8994	. . . . . 6 ⊢ ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
21	19, 20	syl 17	. . . . 5 ⊢ (1o ≺ 𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22		1n0 8500	. . . . . 6 ⊢ 1o ≠ ∅
23		xpsndisj 6152	. . . . . 6 ⊢ (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
24	22, 23	mp1i 13	. . . . 5 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25		undom 9073	. . . . 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 9125	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
28	7, 27	mpdan 687	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {1o}))
29		sdomentr 9125	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
30	17, 29	mpdan 687	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {∅}))
31		unxpdom 9261	. . . . 5 ⊢ ((1o ≺ (𝐴 × {1o}) ∧ 1o ≺ (𝐴 × {∅})) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
32	28, 30, 31	syl2anc 584	. . . 4 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
33		domtr 9021	. . . 4 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
34	26, 32, 33	syl2anc 584	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
35		xpen 9154	. . . 4 ⊢ (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
36	6, 16, 35	syl2anc 584	. . 3 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37		domentr 9027	. . 3 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})) ∧ ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
38	34, 36, 37	syl2anc 584	. 2 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
39	1, 38	eqbrtrid 5154	1 ⊢ (1o ≺ 𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  Vcvv 3459   ∪ cun 3924   ∩ cin 3925  ∅c0 4308  {csn 4601   class class class wbr 5119   × cxp 5652  Oncon0 6352  suc csuc 6354  1_oc1o 8473   ≈ cen 8956   ≼ cdom 8957   ≺ csdm 8958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-1st 7988  df-2nd 7989  df-1o 8480  df-2o 8481  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962

This theorem is referenced by: (None)