Theorem sucxpdom 8711

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 6165	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		relsdom 8499	. . . . . . . . 9 ⊢ Rel ≺
3	2	brrelex2i 5573	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
4		1on 8092	. . . . . . . 8 ⊢ 1o ∈ On
5		xpsneng 8585	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
6	3, 4, 5	sylancl 589	. . . . . . 7 ⊢ (1o ≺ 𝐴 → (𝐴 × {1o}) ≈ 𝐴)
7	6	ensymd 8543	. . . . . 6 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {1o}))
8		endom 8519	. . . . . 6 ⊢ (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
9	7, 8	syl 17	. . . . 5 ⊢ (1o ≺ 𝐴 → 𝐴 ≼ (𝐴 × {1o}))
10		ensn1g 8557	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o)
11	3, 10	syl 17	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → {𝐴} ≈ 1o)
12		ensdomtr 8637	. . . . . . . 8 ⊢ (({𝐴} ≈ 1o ∧ 1o ≺ 𝐴) → {𝐴} ≺ 𝐴)
13	11, 12	mpancom 687	. . . . . . 7 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ 𝐴)
14		0ex 5175	. . . . . . . . 9 ⊢ ∅ ∈ V
15		xpsneng 8585	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
16	3, 14, 15	sylancl 589	. . . . . . . 8 ⊢ (1o ≺ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
17	16	ensymd 8543	. . . . . . 7 ⊢ (1o ≺ 𝐴 → 𝐴 ≈ (𝐴 × {∅}))
18		sdomentr 8635	. . . . . . 7 ⊢ (({𝐴} ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
19	13, 17, 18	syl2anc 587	. . . . . 6 ⊢ (1o ≺ 𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20		sdomdom 8520	. . . . . 6 ⊢ ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
21	19, 20	syl 17	. . . . 5 ⊢ (1o ≺ 𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22		1n0 8102	. . . . . 6 ⊢ 1o ≠ ∅
23		xpsndisj 5987	. . . . . 6 ⊢ (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
24	22, 23	mp1i 13	. . . . 5 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25		undom 8588	. . . . 5 ⊢ (((𝐴 ≼ (𝐴 × {1o}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
26	9, 21, 24, 25	syl21anc 836	. . . 4 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
27		sdomentr 8635	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
28	7, 27	mpdan 686	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {1o}))
29		sdomentr 8635	. . . . . 6 ⊢ ((1o ≺ 𝐴 ∧ 𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
30	17, 29	mpdan 686	. . . . 5 ⊢ (1o ≺ 𝐴 → 1o ≺ (𝐴 × {∅}))
31		unxpdom 8709	. . . . 5 ⊢ ((1o ≺ (𝐴 × {1o}) ∧ 1o ≺ (𝐴 × {∅})) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
32	28, 30, 31	syl2anc 587	. . . 4 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
33		domtr 8545	. . . 4 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
34	26, 32, 33	syl2anc 587	. . 3 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
35		xpen 8664	. . . 4 ⊢ (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
36	6, 16, 35	syl2anc 587	. . 3 ⊢ (1o ≺ 𝐴 → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37		domentr 8551	. . 3 ⊢ (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})) ∧ ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
38	34, 36, 37	syl2anc 587	. 2 ⊢ (1o ≺ 𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
39	1, 38	eqbrtrid 5065	1 ⊢ (1o ≺ 𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441   ∪ cun 3879   ∩ cin 3880  ∅c0 4243  {csn 4525   class class class wbr 5030   × cxp 5517  Oncon0 6159  suc csuc 6161  1_oc1o 8078   ≈ cen 8489   ≼ cdom 8490   ≺ csdm 8491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1st 7671  df-2nd 7672  df-1o 8085  df-2o 8086  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495

This theorem is referenced by: (None)