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Theorem sucxpdom 9257
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 6364 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 8948 . . . . . . . . 9 Rel ≺
32brrelex2i 5726 . . . . . . . 8 (1o𝐴𝐴 ∈ V)
4 1on 8479 . . . . . . . 8 1o ∈ On
5 xpsneng 9058 . . . . . . . 8 ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
63, 4, 5sylancl 585 . . . . . . 7 (1o𝐴 → (𝐴 × {1o}) ≈ 𝐴)
76ensymd 9003 . . . . . 6 (1o𝐴𝐴 ≈ (𝐴 × {1o}))
8 endom 8977 . . . . . 6 (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
97, 8syl 17 . . . . 5 (1o𝐴𝐴 ≼ (𝐴 × {1o}))
10 ensn1g 9021 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1o)
113, 10syl 17 . . . . . . . 8 (1o𝐴 → {𝐴} ≈ 1o)
12 ensdomtr 9115 . . . . . . . 8 (({𝐴} ≈ 1o ∧ 1o𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 685 . . . . . . 7 (1o𝐴 → {𝐴} ≺ 𝐴)
14 0ex 5300 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 9058 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 585 . . . . . . . 8 (1o𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 9003 . . . . . . 7 (1o𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 9113 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 583 . . . . . 6 (1o𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 8978 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1o𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 8489 . . . . . 6 1o ≠ ∅
23 xpsndisj 6156 . . . . . 6 (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1o𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25 undom 9061 . . . . 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 9113 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
287, 27mpdan 684 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {1o}))
29 sdomentr 9113 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
3017, 29mpdan 684 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {∅}))
31 unxpdom 9255 . . . . 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 9005 . . . 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 9142 . . . 4 (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 583 . . 3 (1o𝐴 → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 9011 . . 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 5176 1 (1o𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wne 2934  Vcvv 3468  cun 3941  cin 3942  c0 4317  {csn 4623   class class class wbr 5141   × cxp 5667  Oncon0 6358  suc csuc 6360  1oc1o 8460  cen 8938  cdom 8939  csdm 8940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7974  df-2nd 7975  df-1o 8467  df-2o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944
This theorem is referenced by: (None)
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