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Theorem sucxpdom 8285
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 5842 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 8079 . . . . . . . . 9 Rel ≺
32brrelex2i 5268 . . . . . . . 8 (1𝑜𝐴𝐴 ∈ V)
4 1on 7687 . . . . . . . 8 1𝑜 ∈ On
5 xpsneng 8161 . . . . . . . 8 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
63, 4, 5sylancl 697 . . . . . . 7 (1𝑜𝐴 → (𝐴 × {1𝑜}) ≈ 𝐴)
76ensymd 8123 . . . . . 6 (1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜}))
8 endom 8099 . . . . . 6 (𝐴 ≈ (𝐴 × {1𝑜}) → 𝐴 ≼ (𝐴 × {1𝑜}))
97, 8syl 17 . . . . 5 (1𝑜𝐴𝐴 ≼ (𝐴 × {1𝑜}))
10 ensn1g 8137 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
113, 10syl 17 . . . . . . . 8 (1𝑜𝐴 → {𝐴} ≈ 1𝑜)
12 ensdomtr 8212 . . . . . . . 8 (({𝐴} ≈ 1𝑜 ∧ 1𝑜𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 706 . . . . . . 7 (1𝑜𝐴 → {𝐴} ≺ 𝐴)
14 0ex 4898 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 8161 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 697 . . . . . . . 8 (1𝑜𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 8123 . . . . . . 7 (1𝑜𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 8210 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 696 . . . . . 6 (1𝑜𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 8100 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1𝑜𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 7695 . . . . . 6 1𝑜 ≠ ∅
23 xpsndisj 5667 . . . . . 6 (1𝑜 ≠ ∅ → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1𝑜𝐴 → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
25 undom 8164 . . . . 5 (((𝐴 ≼ (𝐴 × {1𝑜}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
269, 21, 24, 25syl21anc 1438 . . . 4 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
27 sdomentr 8210 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜})) → 1𝑜 ≺ (𝐴 × {1𝑜}))
287, 27mpdan 705 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {1𝑜}))
29 sdomentr 8210 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {∅})) → 1𝑜 ≺ (𝐴 × {∅}))
3017, 29mpdan 705 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
31 unxpdom 8283 . . . . 5 ((1𝑜 ≺ (𝐴 × {1𝑜}) ∧ 1𝑜 ≺ (𝐴 × {∅})) → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
3228, 30, 31syl2anc 696 . . . 4 (1𝑜𝐴 → ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
33 domtr 8125 . . . 4 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
3426, 32, 33syl2anc 696 . . 3 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})))
35 xpen 8239 . . . 4 (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 696 . . 3 (1𝑜𝐴 → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 8131 . . 3 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
3834, 36, 37syl2anc 696 . 2 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
391, 38syl5eqbr 4795 1 (1𝑜𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  wne 2896  Vcvv 3304  cun 3678  cin 3679  c0 4023  {csn 4285   class class class wbr 4760   × cxp 5216  Oncon0 5836  suc csuc 5838  1𝑜c1o 7673  cen 8069  cdom 8070  csdm 8071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-om 7183  df-1st 7285  df-2nd 7286  df-1o 7680  df-2o 7681  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075
This theorem is referenced by: (None)
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