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Theorem isfin4-3 9089
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9071 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
isfin4-3 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))

Proof of Theorem isfin4-3
StepHypRef Expression
1 1on 7519 . . . 4 1𝑜 ∈ On
2 cdadom3 8962 . . . 4 ((𝐴 ∈ FinIV ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
31, 2mpan2 706 . . 3 (𝐴 ∈ FinIV𝐴 ≼ (𝐴 +𝑐 1𝑜))
4 ssun1 3759 . . . . . . . 8 (𝐴 × {∅}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
5 relen 7912 . . . . . . . . . 10 Rel ≈
65brrelexi 5123 . . . . . . . . 9 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
7 cdaval 8944 . . . . . . . . 9 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
86, 1, 7sylancl 693 . . . . . . . 8 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
94, 8syl5sseqr 3638 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊆ (𝐴 +𝑐 1𝑜))
10 0lt1o 7536 . . . . . . . . . 10 ∅ ∈ 1𝑜
111elexi 3202 . . . . . . . . . . 11 1𝑜 ∈ V
1211snid 4184 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
13 opelxpi 5113 . . . . . . . . . 10 ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}))
1410, 12, 13mp2an 707 . . . . . . . . 9 ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})
15 elun2 3764 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1614, 15mp1i 13 . . . . . . . 8 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1716, 8eleqtrrd 2701 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
18 1n0 7527 . . . . . . . 8 1𝑜 ≠ ∅
19 opelxp2 5116 . . . . . . . . . 10 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 ∈ {∅})
20 elsni 4170 . . . . . . . . . 10 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
2119, 20syl 17 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 = ∅)
2221necon3ai 2815 . . . . . . . 8 (1𝑜 ≠ ∅ → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
2318, 22mp1i 13 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
249, 17, 23ssnelpssd 3702 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜))
25 0ex 4755 . . . . . . . 8 ∅ ∈ V
26 xpsneng 7997 . . . . . . . 8 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
276, 25, 26sylancl 693 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
28 entr 7960 . . . . . . 7 (((𝐴 × {∅}) ≈ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜)) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
2927, 28mpancom 702 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
30 fin4i 9072 . . . . . 6 (((𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜) ∧ (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜)) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
3124, 29, 30syl2anc 692 . . . . 5 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
32 fin4en1 9083 . . . . 5 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV → (𝐴 +𝑐 1𝑜) ∈ FinIV))
3331, 32mtod 189 . . . 4 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ∈ FinIV)
3433con2i 134 . . 3 (𝐴 ∈ FinIV → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
35 brsdom 7930 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) ↔ (𝐴 ≼ (𝐴 +𝑐 1𝑜) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜)))
363, 34, 35sylanbrc 697 . 2 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))
37 sdomnen 7936 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
38 infcda1 8967 . . . . 5 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
3938ensymd 7959 . . . 4 (ω ≼ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜))
4037, 39nsyl 135 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ ω ≼ 𝐴)
41 relsdom 7914 . . . . 5 Rel ≺
4241brrelexi 5123 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
43 isfin4-2 9088 . . . 4 (𝐴 ∈ V → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
4442, 43syl 17 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
4540, 44mpbird 247 . 2 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ FinIV)
4636, 45impbii 199 1 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1480  wcel 1987  wne 2790  Vcvv 3189  cun 3557  wpss 3560  c0 3896  {csn 4153  cop 4159   class class class wbr 4618   × cxp 5077  Oncon0 5687  (class class class)co 6610  ωcom 7019  1𝑜c1o 7505  cen 7904  cdom 7905  csdm 7906   +𝑐 ccda 8941  FinIVcfin4 9054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-cda 8942  df-fin4 9061
This theorem is referenced by:  fin45  9166  finngch  9429  gchinf  9431
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