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Theorem sucxpdom 8336
 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 5890 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 8130 . . . . . . . . 9 Rel ≺
32brrelex2i 5316 . . . . . . . 8 (1𝑜𝐴𝐴 ∈ V)
4 1on 7737 . . . . . . . 8 1𝑜 ∈ On
5 xpsneng 8212 . . . . . . . 8 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
63, 4, 5sylancl 697 . . . . . . 7 (1𝑜𝐴 → (𝐴 × {1𝑜}) ≈ 𝐴)
76ensymd 8174 . . . . . 6 (1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜}))
8 endom 8150 . . . . . 6 (𝐴 ≈ (𝐴 × {1𝑜}) → 𝐴 ≼ (𝐴 × {1𝑜}))
97, 8syl 17 . . . . 5 (1𝑜𝐴𝐴 ≼ (𝐴 × {1𝑜}))
10 ensn1g 8188 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
113, 10syl 17 . . . . . . . 8 (1𝑜𝐴 → {𝐴} ≈ 1𝑜)
12 ensdomtr 8263 . . . . . . . 8 (({𝐴} ≈ 1𝑜 ∧ 1𝑜𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 706 . . . . . . 7 (1𝑜𝐴 → {𝐴} ≺ 𝐴)
14 0ex 4942 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 8212 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 697 . . . . . . . 8 (1𝑜𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 8174 . . . . . . 7 (1𝑜𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 8261 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 696 . . . . . 6 (1𝑜𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 8151 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1𝑜𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 7746 . . . . . 6 1𝑜 ≠ ∅
23 xpsndisj 5715 . . . . . 6 (1𝑜 ≠ ∅ → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1𝑜𝐴 → ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅)
25 undom 8215 . . . . 5 (((𝐴 ≼ (𝐴 × {1𝑜}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1𝑜}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
269, 21, 24, 25syl21anc 1476 . . . 4 (1𝑜𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1𝑜}) ∪ (𝐴 × {∅})))
27 sdomentr 8261 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {1𝑜})) → 1𝑜 ≺ (𝐴 × {1𝑜}))
287, 27mpdan 705 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {1𝑜}))
29 sdomentr 8261 . . . . . 6 ((1𝑜𝐴𝐴 ≈ (𝐴 × {∅})) → 1𝑜 ≺ (𝐴 × {∅}))
3017, 29mpdan 705 . . . . 5 (1𝑜𝐴 → 1𝑜 ≺ (𝐴 × {∅}))
31 unxpdom 8334 . . . . 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 8176 . . . 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 8290 . . . 4 (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 696 . . 3 (1𝑜𝐴 → ((𝐴 × {1𝑜}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 8182 . . 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 4839 1 (1𝑜𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  Vcvv 3340   ∪ cun 3713   ∩ cin 3714  ∅c0 4058  {csn 4321   class class class wbr 4804   × cxp 5264  Oncon0 5884  suc csuc 5886  1𝑜c1o 7723   ≈ cen 8120   ≼ cdom 8121   ≺ csdm 8122 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-1st 7334  df-2nd 7335  df-1o 7730  df-2o 7731  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126 This theorem is referenced by: (None)
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