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Theorem sucxpdom 8375
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 5913 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 8166 . . . . . . . . 9 Rel ≺
32brrelex2i 5328 . . . . . . . 8 (1𝑜𝐴𝐴 ∈ V)
4 1on 7770 . . . . . . . 8 1𝑜 ∈ On
5 xpsneng 8251 . . . . . . . 8 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
63, 4, 5sylancl 580 . . . . . . 7 (1𝑜𝐴 → (𝐴 × {1𝑜}) ≈ 𝐴)
76ensymd 8210 . . . . . 6 (1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜}))
8 endom 8186 . . . . . 6 (𝐴 ≈ (𝐴 × {1𝑜}) → 𝐴 ≼ (𝐴 × {1𝑜}))
97, 8syl 17 . . . . 5 (1𝑜𝐴𝐴 ≼ (𝐴 × {1𝑜}))
10 ensn1g 8224 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
113, 10syl 17 . . . . . . . 8 (1𝑜𝐴 → {𝐴} ≈ 1𝑜)
12 ensdomtr 8302 . . . . . . . 8 (({𝐴} ≈ 1𝑜 ∧ 1𝑜𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 679 . . . . . . 7 (1𝑜𝐴 → {𝐴} ≺ 𝐴)
14 0ex 4949 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 8251 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 580 . . . . . . . 8 (1𝑜𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 8210 . . . . . . 7 (1𝑜𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 8300 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 579 . . . . . 6 (1𝑜𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 8187 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1𝑜𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 7779 . . . . . 6 1𝑜 ≠ ∅
23 xpsndisj 5739 . . . . . 6 (1𝑜 ≠ ∅ → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1𝑜𝐴 → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
25 undom 8254 . . . . 5 (((𝐴 ≼ (𝐴 × {1𝑜}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
269, 21, 24, 25syl21anc 866 . . . 4 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
27 sdomentr 8300 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜})) → 1𝑜 ≺ (𝐴 × {1𝑜}))
287, 27mpdan 678 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {1𝑜}))
29 sdomentr 8300 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {∅})) → 1𝑜 ≺ (𝐴 × {∅}))
3017, 29mpdan 678 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
31 unxpdom 8373 . . . . 5 ((1𝑜 ≺ (𝐴 × {1𝑜}) ∧ 1𝑜 ≺ (𝐴 × {∅})) → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
3228, 30, 31syl2anc 579 . . . 4 (1𝑜𝐴 → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
33 domtr 8212 . . . 4 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
3426, 32, 33syl2anc 579 . . 3 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
35 xpen 8329 . . . 4 (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 579 . . 3 (1𝑜𝐴 → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 8218 . . 3 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
3834, 36, 37syl2anc 579 . 2 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
391, 38syl5eqbr 4843 1 (1𝑜𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  wne 2936  Vcvv 3349  cun 3729  cin 3730  c0 4078  {csn 4333   class class class wbr 4808   × cxp 5274  Oncon0 5907  suc csuc 5909  1𝑜c1o 7756  cen 8156  cdom 8157  csdm 8158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-om 7263  df-1st 7365  df-2nd 7366  df-1o 7763  df-2o 7764  df-er 7946  df-en 8160  df-dom 8161  df-sdom 8162
This theorem is referenced by: (None)
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