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Theorem sucxpdom 8027
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 5628 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 7821 . . . . . . . . 9 Rel ≺
32brrelex2i 5069 . . . . . . . 8 (1𝑜𝐴𝐴 ∈ V)
4 1on 7427 . . . . . . . 8 1𝑜 ∈ On
5 xpsneng 7903 . . . . . . . 8 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
63, 4, 5sylancl 692 . . . . . . 7 (1𝑜𝐴 → (𝐴 × {1𝑜}) ≈ 𝐴)
76ensymd 7866 . . . . . 6 (1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜}))
8 endom 7841 . . . . . 6 (𝐴 ≈ (𝐴 × {1𝑜}) → 𝐴 ≼ (𝐴 × {1𝑜}))
97, 8syl 17 . . . . 5 (1𝑜𝐴𝐴 ≼ (𝐴 × {1𝑜}))
10 ensn1g 7880 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
113, 10syl 17 . . . . . . . 8 (1𝑜𝐴 → {𝐴} ≈ 1𝑜)
12 ensdomtr 7954 . . . . . . . 8 (({𝐴} ≈ 1𝑜 ∧ 1𝑜𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 699 . . . . . . 7 (1𝑜𝐴 → {𝐴} ≺ 𝐴)
14 0ex 4709 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 7903 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 692 . . . . . . . 8 (1𝑜𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 7866 . . . . . . 7 (1𝑜𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 7952 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 690 . . . . . 6 (1𝑜𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 7842 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1𝑜𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 7435 . . . . . 6 1𝑜 ≠ ∅
23 xpsndisj 5458 . . . . . 6 (1𝑜 ≠ ∅ → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1𝑜𝐴 → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
25 undom 7906 . . . . 5 (((𝐴 ≼ (𝐴 × {1𝑜}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
269, 21, 24, 25syl21anc 1316 . . . 4 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
27 sdomentr 7952 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜})) → 1𝑜 ≺ (𝐴 × {1𝑜}))
287, 27mpdan 698 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {1𝑜}))
29 sdomentr 7952 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {∅})) → 1𝑜 ≺ (𝐴 × {∅}))
3017, 29mpdan 698 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
31 unxpdom 8025 . . . . 5 ((1𝑜 ≺ (𝐴 × {1𝑜}) ∧ 1𝑜 ≺ (𝐴 × {∅})) → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
3228, 30, 31syl2anc 690 . . . 4 (1𝑜𝐴 → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
33 domtr 7868 . . . 4 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
3426, 32, 33syl2anc 690 . . 3 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
35 xpen 7981 . . . 4 (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 690 . . 3 (1𝑜𝐴 → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 7874 . . 3 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
3834, 36, 37syl2anc 690 . 2 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
391, 38syl5eqbr 4608 1 (1𝑜𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  wne 2775  Vcvv 3168  cun 3533  cin 3534  c0 3869  {csn 4120   class class class wbr 4573   × cxp 5022  Oncon0 5622  suc csuc 5624  1𝑜c1o 7413  cen 7811  cdom 7812  csdm 7813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-om 6931  df-1st 7032  df-2nd 7033  df-1o 7420  df-2o 7421  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817
This theorem is referenced by: (None)
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