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Theorem sucxpdom 9291
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 (1o𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

Proof of Theorem sucxpdom
StepHypRef Expression
1 df-suc 6390 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 8992 . . . . . . . . 9 Rel ≺
32brrelex2i 5742 . . . . . . . 8 (1o𝐴𝐴 ∈ V)
4 1on 8518 . . . . . . . 8 1o ∈ On
5 xpsneng 9096 . . . . . . . 8 ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
63, 4, 5sylancl 586 . . . . . . 7 (1o𝐴 → (𝐴 × {1o}) ≈ 𝐴)
76ensymd 9045 . . . . . 6 (1o𝐴𝐴 ≈ (𝐴 × {1o}))
8 endom 9019 . . . . . 6 (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
97, 8syl 17 . . . . 5 (1o𝐴𝐴 ≼ (𝐴 × {1o}))
10 ensn1g 9062 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1o)
113, 10syl 17 . . . . . . . 8 (1o𝐴 → {𝐴} ≈ 1o)
12 ensdomtr 9153 . . . . . . . 8 (({𝐴} ≈ 1o ∧ 1o𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 688 . . . . . . 7 (1o𝐴 → {𝐴} ≺ 𝐴)
14 0ex 5307 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 9096 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 586 . . . . . . . 8 (1o𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 9045 . . . . . . 7 (1o𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 9151 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 584 . . . . . 6 (1o𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 9020 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1o𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 8526 . . . . . 6 1o ≠ ∅
23 xpsndisj 6183 . . . . . 6 (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1o𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25 undom 9099 . . . . 5 (((𝐴 ≼ (𝐴 × {1o}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
269, 21, 24, 25syl21anc 838 . . . 4 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
27 sdomentr 9151 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
287, 27mpdan 687 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {1o}))
29 sdomentr 9151 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
3017, 29mpdan 687 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {∅}))
31 unxpdom 9289 . . . . 5 ((1o ≺ (𝐴 × {1o}) ∧ 1o ≺ (𝐴 × {∅})) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
3228, 30, 31syl2anc 584 . . . 4 (1o𝐴 → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
33 domtr 9047 . . . 4 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
3426, 32, 33syl2anc 584 . . 3 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
35 xpen 9180 . . . 4 (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 584 . . 3 (1o𝐴 → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 9053 . . 3 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})) ∧ ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
3834, 36, 37syl2anc 584 . 2 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
391, 38eqbrtrid 5178 1 (1o𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cun 3949  cin 3950  c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  Oncon0 6384  suc csuc 6386  1oc1o 8499  cen 8982  cdom 8983  csdm 8984
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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988
This theorem is referenced by: (None)
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