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Theorem sucxpdom 9261
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 6370 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 8952 . . . . . . . . 9 Rel ≺
32brrelex2i 5733 . . . . . . . 8 (1o𝐴𝐴 ∈ V)
4 1on 8484 . . . . . . . 8 1o ∈ On
5 xpsneng 9062 . . . . . . . 8 ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
63, 4, 5sylancl 585 . . . . . . 7 (1o𝐴 → (𝐴 × {1o}) ≈ 𝐴)
76ensymd 9007 . . . . . 6 (1o𝐴𝐴 ≈ (𝐴 × {1o}))
8 endom 8981 . . . . . 6 (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
97, 8syl 17 . . . . 5 (1o𝐴𝐴 ≼ (𝐴 × {1o}))
10 ensn1g 9025 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1o)
113, 10syl 17 . . . . . . . 8 (1o𝐴 → {𝐴} ≈ 1o)
12 ensdomtr 9119 . . . . . . . 8 (({𝐴} ≈ 1o ∧ 1o𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 685 . . . . . . 7 (1o𝐴 → {𝐴} ≺ 𝐴)
14 0ex 5307 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 9062 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 585 . . . . . . . 8 (1o𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 9007 . . . . . . 7 (1o𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 9117 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 583 . . . . . 6 (1o𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 8982 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1o𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 8494 . . . . . 6 1o ≠ ∅
23 xpsndisj 6162 . . . . . 6 (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1o𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25 undom 9065 . . . . 5 (((𝐴 ≼ (𝐴 × {1o}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
269, 21, 24, 25syl21anc 835 . . . 4 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
27 sdomentr 9117 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
287, 27mpdan 684 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {1o}))
29 sdomentr 9117 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
3017, 29mpdan 684 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {∅}))
31 unxpdom 9259 . . . . 5 ((1o ≺ (𝐴 × {1o}) ∧ 1o ≺ (𝐴 × {∅})) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
3228, 30, 31syl2anc 583 . . . 4 (1o𝐴 → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
33 domtr 9009 . . . 4 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
3426, 32, 33syl2anc 583 . . 3 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
35 xpen 9146 . . . 4 (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 583 . . 3 (1o𝐴 → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 9015 . . 3 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})) ∧ ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
3834, 36, 37syl2anc 583 . 2 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
391, 38eqbrtrid 5183 1 (1o𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  wne 2939  Vcvv 3473  cun 3946  cin 3947  c0 4322  {csn 4628   class class class wbr 5148   × cxp 5674  Oncon0 6364  suc csuc 6366  1oc1o 8465  cen 8942  cdom 8943  csdm 8944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 7979  df-2nd 7980  df-1o 8472  df-2o 8473  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948
This theorem is referenced by: (None)
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