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Theorem sucxpdom 9288
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 6380 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 8979 . . . . . . . . 9 Rel ≺
32brrelex2i 5739 . . . . . . . 8 (1o𝐴𝐴 ∈ V)
4 1on 8507 . . . . . . . 8 1o ∈ On
5 xpsneng 9089 . . . . . . . 8 ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
63, 4, 5sylancl 584 . . . . . . 7 (1o𝐴 → (𝐴 × {1o}) ≈ 𝐴)
76ensymd 9034 . . . . . 6 (1o𝐴𝐴 ≈ (𝐴 × {1o}))
8 endom 9008 . . . . . 6 (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
97, 8syl 17 . . . . 5 (1o𝐴𝐴 ≼ (𝐴 × {1o}))
10 ensn1g 9052 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1o)
113, 10syl 17 . . . . . . . 8 (1o𝐴 → {𝐴} ≈ 1o)
12 ensdomtr 9146 . . . . . . . 8 (({𝐴} ≈ 1o ∧ 1o𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 686 . . . . . . 7 (1o𝐴 → {𝐴} ≺ 𝐴)
14 0ex 5311 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 9089 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 584 . . . . . . . 8 (1o𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 9034 . . . . . . 7 (1o𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 9144 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 582 . . . . . 6 (1o𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 9009 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1o𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 8517 . . . . . 6 1o ≠ ∅
23 xpsndisj 6172 . . . . . 6 (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1o𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25 undom 9092 . . . . 5 (((𝐴 ≼ (𝐴 × {1o}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
269, 21, 24, 25syl21anc 836 . . . 4 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
27 sdomentr 9144 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
287, 27mpdan 685 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {1o}))
29 sdomentr 9144 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
3017, 29mpdan 685 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {∅}))
31 unxpdom 9286 . . . . 5 ((1o ≺ (𝐴 × {1o}) ∧ 1o ≺ (𝐴 × {∅})) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
3228, 30, 31syl2anc 582 . . . 4 (1o𝐴 → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
33 domtr 9036 . . . 4 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
3426, 32, 33syl2anc 582 . . 3 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
35 xpen 9173 . . . 4 (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 582 . . 3 (1o𝐴 → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 9042 . . 3 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})) ∧ ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
3834, 36, 37syl2anc 582 . 2 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
391, 38eqbrtrid 5187 1 (1o𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wne 2937  Vcvv 3473  cun 3947  cin 3948  c0 4326  {csn 4632   class class class wbr 5152   × cxp 5680  Oncon0 6374  suc csuc 6376  1oc1o 8488  cen 8969  cdom 8970  csdm 8971
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1st 8001  df-2nd 8002  df-1o 8495  df-2o 8496  df-er 8733  df-en 8973  df-dom 8974  df-sdom 8975
This theorem is referenced by: (None)
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