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Theorem sucxpdom 9206
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 6328 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 relsdom 8897 . . . . . . . . 9 Rel ≺
32brrelex2i 5694 . . . . . . . 8 (1o𝐴𝐴 ∈ V)
4 1on 8429 . . . . . . . 8 1o ∈ On
5 xpsneng 9007 . . . . . . . 8 ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
63, 4, 5sylancl 587 . . . . . . 7 (1o𝐴 → (𝐴 × {1o}) ≈ 𝐴)
76ensymd 8952 . . . . . 6 (1o𝐴𝐴 ≈ (𝐴 × {1o}))
8 endom 8926 . . . . . 6 (𝐴 ≈ (𝐴 × {1o}) → 𝐴 ≼ (𝐴 × {1o}))
97, 8syl 17 . . . . 5 (1o𝐴𝐴 ≼ (𝐴 × {1o}))
10 ensn1g 8970 . . . . . . . . 9 (𝐴 ∈ V → {𝐴} ≈ 1o)
113, 10syl 17 . . . . . . . 8 (1o𝐴 → {𝐴} ≈ 1o)
12 ensdomtr 9064 . . . . . . . 8 (({𝐴} ≈ 1o ∧ 1o𝐴) → {𝐴} ≺ 𝐴)
1311, 12mpancom 687 . . . . . . 7 (1o𝐴 → {𝐴} ≺ 𝐴)
14 0ex 5269 . . . . . . . . 9 ∅ ∈ V
15 xpsneng 9007 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
163, 14, 15sylancl 587 . . . . . . . 8 (1o𝐴 → (𝐴 × {∅}) ≈ 𝐴)
1716ensymd 8952 . . . . . . 7 (1o𝐴𝐴 ≈ (𝐴 × {∅}))
18 sdomentr 9062 . . . . . . 7 (({𝐴} ≺ 𝐴𝐴 ≈ (𝐴 × {∅})) → {𝐴} ≺ (𝐴 × {∅}))
1913, 17, 18syl2anc 585 . . . . . 6 (1o𝐴 → {𝐴} ≺ (𝐴 × {∅}))
20 sdomdom 8927 . . . . . 6 ({𝐴} ≺ (𝐴 × {∅}) → {𝐴} ≼ (𝐴 × {∅}))
2119, 20syl 17 . . . . 5 (1o𝐴 → {𝐴} ≼ (𝐴 × {∅}))
22 1n0 8439 . . . . . 6 1o ≠ ∅
23 xpsndisj 6120 . . . . . 6 (1o ≠ ∅ → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
2422, 23mp1i 13 . . . . 5 (1o𝐴 → ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅)
25 undom 9010 . . . . 5 (((𝐴 ≼ (𝐴 × {1o}) ∧ {𝐴} ≼ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∩ (𝐴 × {∅})) = ∅) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
269, 21, 24, 25syl21anc 837 . . . 4 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})))
27 sdomentr 9062 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {1o})) → 1o ≺ (𝐴 × {1o}))
287, 27mpdan 686 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {1o}))
29 sdomentr 9062 . . . . . 6 ((1o𝐴𝐴 ≈ (𝐴 × {∅})) → 1o ≺ (𝐴 × {∅}))
3017, 29mpdan 686 . . . . 5 (1o𝐴 → 1o ≺ (𝐴 × {∅}))
31 unxpdom 9204 . . . . 5 ((1o ≺ (𝐴 × {1o}) ∧ 1o ≺ (𝐴 × {∅})) → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
3228, 30, 31syl2anc 585 . . . 4 (1o𝐴 → ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
33 domtr 8954 . . . 4 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ∧ ((𝐴 × {1o}) ∪ (𝐴 × {∅})) ≼ ((𝐴 × {1o}) × (𝐴 × {∅}))) → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
3426, 32, 33syl2anc 585 . . 3 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})))
35 xpen 9091 . . . 4 (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐴 × {∅}) ≈ 𝐴) → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
366, 16, 35syl2anc 585 . . 3 (1o𝐴 → ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴))
37 domentr 8960 . . 3 (((𝐴 ∪ {𝐴}) ≼ ((𝐴 × {1o}) × (𝐴 × {∅})) ∧ ((𝐴 × {1o}) × (𝐴 × {∅})) ≈ (𝐴 × 𝐴)) → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
3834, 36, 37syl2anc 585 . 2 (1o𝐴 → (𝐴 ∪ {𝐴}) ≼ (𝐴 × 𝐴))
391, 38eqbrtrid 5145 1 (1o𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wne 2944  Vcvv 3448  cun 3913  cin 3914  c0 4287  {csn 4591   class class class wbr 5110   × cxp 5636  Oncon0 6322  suc csuc 6324  1oc1o 8410  cen 8887  cdom 8888  csdm 8889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1st 7926  df-2nd 7927  df-1o 8417  df-2o 8418  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893
This theorem is referenced by: (None)
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