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Theorem isfin4-3 9134
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9116 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 7564 . . . 4 1𝑜 ∈ On
2 cdadom3 9007 . . . 4 ((𝐴 ∈ FinIV ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
31, 2mpan2 707 . . 3 (𝐴 ∈ FinIV𝐴 ≼ (𝐴 +𝑐 1𝑜))
4 ssun1 3774 . . . . . . . 8 (𝐴 × {∅}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
5 relen 7957 . . . . . . . . . 10 Rel ≈
65brrelexi 5156 . . . . . . . . 9 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
7 cdaval 8989 . . . . . . . . 9 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
86, 1, 7sylancl 694 . . . . . . . 8 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
94, 8syl5sseqr 3652 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊆ (𝐴 +𝑐 1𝑜))
10 0lt1o 7581 . . . . . . . . . 10 ∅ ∈ 1𝑜
111elexi 3211 . . . . . . . . . . 11 1𝑜 ∈ V
1211snid 4206 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
13 opelxpi 5146 . . . . . . . . . 10 ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}))
1410, 12, 13mp2an 708 . . . . . . . . 9 ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})
15 elun2 3779 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1614, 15mp1i 13 . . . . . . . 8 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1716, 8eleqtrrd 2703 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
18 1n0 7572 . . . . . . . 8 1𝑜 ≠ ∅
19 opelxp2 5149 . . . . . . . . . 10 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 ∈ {∅})
20 elsni 4192 . . . . . . . . . 10 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
2119, 20syl 17 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 = ∅)
2221necon3ai 2818 . . . . . . . 8 (1𝑜 ≠ ∅ → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
2318, 22mp1i 13 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
249, 17, 23ssnelpssd 3717 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜))
25 0ex 4788 . . . . . . . 8 ∅ ∈ V
26 xpsneng 8042 . . . . . . . 8 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
276, 25, 26sylancl 694 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
28 entr 8005 . . . . . . 7 (((𝐴 × {∅}) ≈ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜)) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
2927, 28mpancom 703 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
30 fin4i 9117 . . . . . 6 (((𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜) ∧ (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜)) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
3124, 29, 30syl2anc 693 . . . . 5 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
32 fin4en1 9128 . . . . 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 7975 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) ↔ (𝐴 ≼ (𝐴 +𝑐 1𝑜) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜)))
363, 34, 35sylanbrc 698 . 2 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))
37 sdomnen 7981 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
38 infcda1 9012 . . . . 5 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
3938ensymd 8004 . . . 4 (ω ≼ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜))
4037, 39nsyl 135 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ ω ≼ 𝐴)
41 relsdom 7959 . . . . 5 Rel ≺
4241brrelexi 5156 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
43 isfin4-2 9133 . . . 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 1482  wcel 1989  wne 2793  Vcvv 3198  cun 3570  wpss 3573  c0 3913  {csn 4175  cop 4181   class class class wbr 4651   × cxp 5110  Oncon0 5721  (class class class)co 6647  ωcom 7062  1𝑜c1o 7550  cen 7949  cdom 7950  csdm 7951   +𝑐 ccda 8986  FinIVcfin4 9099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-cda 8987  df-fin4 9106
This theorem is referenced by:  fin45  9211  finngch  9474  gchinf  9476
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