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Theorem isfin4-3 9423
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9405 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 7804 . . . 4 1𝑜 ∈ On
2 cdadom3 9296 . . . 4 ((𝐴 ∈ FinIV ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
31, 2mpan2 683 . . 3 (𝐴 ∈ FinIV𝐴 ≼ (𝐴 +𝑐 1𝑜))
4 ssun1 3972 . . . . . . . 8 (𝐴 × {∅}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
5 relen 8198 . . . . . . . . . 10 Rel ≈
65brrelex1i 5361 . . . . . . . . 9 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
7 cdaval 9278 . . . . . . . . 9 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
86, 1, 7sylancl 581 . . . . . . . 8 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
94, 8syl5sseqr 3848 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊆ (𝐴 +𝑐 1𝑜))
10 0lt1o 7822 . . . . . . . . . 10 ∅ ∈ 1𝑜
11 1oex 7805 . . . . . . . . . . 11 1𝑜 ∈ V
1211snid 4398 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
13 opelxpi 5347 . . . . . . . . . 10 ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}))
1410, 12, 13mp2an 684 . . . . . . . . 9 ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})
15 elun2 3977 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1614, 15mp1i 13 . . . . . . . 8 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1716, 8eleqtrrd 2879 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
18 1n0 7813 . . . . . . . 8 1𝑜 ≠ ∅
19 opelxp2 5352 . . . . . . . . . 10 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 ∈ {∅})
20 elsni 4383 . . . . . . . . . 10 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
2119, 20syl 17 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 = ∅)
2221necon3ai 2994 . . . . . . . 8 (1𝑜 ≠ ∅ → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
2318, 22mp1i 13 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
249, 17, 23ssnelpssd 3914 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜))
25 0ex 4982 . . . . . . . 8 ∅ ∈ V
26 xpsneng 8285 . . . . . . . 8 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
276, 25, 26sylancl 581 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
28 entr 8245 . . . . . . 7 (((𝐴 × {∅}) ≈ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜)) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
2927, 28mpancom 680 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
30 fin4i 9406 . . . . . 6 (((𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜) ∧ (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜)) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
3124, 29, 30syl2anc 580 . . . . 5 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
32 fin4en1 9417 . . . . 5 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV → (𝐴 +𝑐 1𝑜) ∈ FinIV))
3331, 32mtod 190 . . . 4 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ∈ FinIV)
3433con2i 137 . . 3 (𝐴 ∈ FinIV → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
35 brsdom 8216 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) ↔ (𝐴 ≼ (𝐴 +𝑐 1𝑜) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜)))
363, 34, 35sylanbrc 579 . 2 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))
37 sdomnen 8222 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
38 infcda1 9301 . . . . 5 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
3938ensymd 8244 . . . 4 (ω ≼ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜))
4037, 39nsyl 138 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ ω ≼ 𝐴)
41 relsdom 8200 . . . . 5 Rel ≺
4241brrelex1i 5361 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
43 isfin4-2 9422 . . . 4 (𝐴 ∈ V → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
4442, 43syl 17 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
4540, 44mpbird 249 . 2 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ FinIV)
4636, 45impbii 201 1 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198   = wceq 1653  wcel 2157  wne 2969  Vcvv 3383  cun 3765  wpss 3768  c0 4113  {csn 4366  cop 4372   class class class wbr 4841   × cxp 5308  Oncon0 5939  (class class class)co 6876  ωcom 7297  1𝑜c1o 7790  cen 8190  cdom 8191  csdm 8192   +𝑐 ccda 9275  FinIVcfin4 9388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-er 7980  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-cda 9276  df-fin4 9395
This theorem is referenced by:  fin45  9500  finngch  9763  gchinf  9765
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