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Theorem isfin4-3 9020
 Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9002 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 7454 . . . 4 1𝑜 ∈ On
2 cdadom3 8893 . . . 4 ((𝐴 ∈ FinIV ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
31, 2mpan2 703 . . 3 (𝐴 ∈ FinIV𝐴 ≼ (𝐴 +𝑐 1𝑜))
4 ssun1 3738 . . . . . . . 8 (𝐴 × {∅}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
5 relen 7846 . . . . . . . . . 10 Rel ≈
65brrelexi 5082 . . . . . . . . 9 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
7 cdaval 8875 . . . . . . . . 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 3617 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊆ (𝐴 +𝑐 1𝑜))
10 0lt1o 7471 . . . . . . . . . 10 ∅ ∈ 1𝑜
111elexi 3186 . . . . . . . . . . 11 1𝑜 ∈ V
1211snid 4155 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
13 opelxpi 5072 . . . . . . . . . 10 ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}))
1410, 12, 13mp2an 704 . . . . . . . . 9 ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})
15 elun2 3743 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1614, 15mp1i 13 . . . . . . . 8 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1716, 8eleqtrrd 2691 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
18 1n0 7462 . . . . . . . 8 1𝑜 ≠ ∅
19 opelxp2 5075 . . . . . . . . . 10 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 ∈ {∅})
20 elsni 4142 . . . . . . . . . 10 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
2119, 20syl 17 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 = ∅)
2221necon3ai 2807 . . . . . . . 8 (1𝑜 ≠ ∅ → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
2318, 22mp1i 13 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
249, 17, 23ssnelpssd 3681 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜))
25 0ex 4718 . . . . . . . 8 ∅ ∈ V
26 xpsneng 7930 . . . . . . . 8 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
276, 25, 26sylancl 693 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
28 entr 7894 . . . . . . 7 (((𝐴 × {∅}) ≈ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜)) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
2927, 28mpancom 700 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
30 fin4i 9003 . . . . . 6 (((𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜) ∧ (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜)) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
3124, 29, 30syl2anc 691 . . . . 5 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
32 fin4en1 9014 . . . . 5 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV → (𝐴 +𝑐 1𝑜) ∈ FinIV))
3331, 32mtod 188 . . . 4 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ∈ FinIV)
3433con2i 133 . . 3 (𝐴 ∈ FinIV → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
35 brsdom 7864 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) ↔ (𝐴 ≼ (𝐴 +𝑐 1𝑜) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜)))
363, 34, 35sylanbrc 695 . 2 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))
37 sdomnen 7870 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
38 infcda1 8898 . . . . 5 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
3938ensymd 7893 . . . 4 (ω ≼ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜))
4037, 39nsyl 134 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ ω ≼ 𝐴)
41 relsdom 7848 . . . . 5 Rel ≺
4241brrelexi 5082 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
43 isfin4-2 9019 . . . 4 (𝐴 ∈ V → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
4442, 43syl 17 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
4540, 44mpbird 246 . 2 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ FinIV)
4636, 45impbii 198 1 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∪ cun 3538   ⊊ wpss 3541  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  Oncon0 5640  (class class class)co 6549  ωcom 6957  1𝑜c1o 7440   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840   +𝑐 ccda 8872  FinIVcfin4 8985 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-cda 8873  df-fin4 8992 This theorem is referenced by:  fin45  9097  finngch  9356  gchinf  9358
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