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Theorem sucxpdom 8054
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 5646 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 7848 . . . . . . . . 9 Rel ≺
32brrelex2i 5083 . . . . . . . 8 (1𝑜𝐴𝐴 ∈ V)
4 1on 7454 . . . . . . . 8 1𝑜 ∈ On
5 xpsneng 7930 . . . . . . . 8 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
63, 4, 5sylancl 693 . . . . . . 7 (1𝑜𝐴 → (𝐴 × {1𝑜}) ≈ 𝐴)
76ensymd 7893 . . . . . 6 (1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜}))
8 endom 7868 . . . . . 6 (𝐴 ≈ (𝐴 × {1𝑜}) → 𝐴 ≼ (𝐴 × {1𝑜}))
97, 8syl 17 . . . . 5 (1𝑜𝐴𝐴 ≼ (𝐴 × {1𝑜}))
10 ensn1g 7907 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
113, 10syl 17 . . . . . . . 8 (1𝑜𝐴 → {𝐴} ≈ 1𝑜)
12 ensdomtr 7981 . . . . . . . 8 (({𝐴} ≈ 1𝑜 ∧ 1𝑜𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 700 . . . . . . 7 (1𝑜𝐴 → {𝐴} ≺ 𝐴)
14 0ex 4718 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 7930 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 693 . . . . . . . 8 (1𝑜𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 7893 . . . . . . 7 (1𝑜𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 7979 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 691 . . . . . 6 (1𝑜𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 7869 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1𝑜𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 7462 . . . . . 6 1𝑜 ≠ ∅
23 xpsndisj 5476 . . . . . 6 (1𝑜 ≠ ∅ → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1𝑜𝐴 → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
25 undom 7933 . . . . 5 (((𝐴 ≼ (𝐴 × {1𝑜}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
269, 21, 24, 25syl21anc 1317 . . . 4 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
27 sdomentr 7979 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜})) → 1𝑜 ≺ (𝐴 × {1𝑜}))
287, 27mpdan 699 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {1𝑜}))
29 sdomentr 7979 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {∅})) → 1𝑜 ≺ (𝐴 × {∅}))
3017, 29mpdan 699 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
31 unxpdom 8052 . . . . 5 ((1𝑜 ≺ (𝐴 × {1𝑜}) ∧ 1𝑜 ≺ (𝐴 × {∅})) → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
3228, 30, 31syl2anc 691 . . . 4 (1𝑜𝐴 → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
33 domtr 7895 . . . 4 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
3426, 32, 33syl2anc 691 . . 3 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
35 xpen 8008 . . . 4 (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 691 . . 3 (1𝑜𝐴 → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 7901 . . 3 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
3834, 36, 37syl2anc 691 . 2 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
391, 38syl5eqbr 4618 1 (1𝑜𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  cun 3538  cin 3539  c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  Oncon0 5640  suc csuc 5642  1𝑜c1o 7440  cen 7838  cdom 7839  csdm 7840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844
This theorem is referenced by: (None)
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