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Theorem isfin4-3 9340
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9322 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 7721 . . . 4 1𝑜 ∈ On
2 cdadom3 9213 . . . 4 ((𝐴 ∈ FinIV ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
31, 2mpan2 665 . . 3 (𝐴 ∈ FinIV𝐴 ≼ (𝐴 +𝑐 1𝑜))
4 ssun1 3928 . . . . . . . 8 (𝐴 × {∅}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
5 relen 8115 . . . . . . . . . 10 Rel ≈
65brrelexi 5299 . . . . . . . . 9 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
7 cdaval 9195 . . . . . . . . 9 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
86, 1, 7sylancl 568 . . . . . . . 8 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
94, 8syl5sseqr 3804 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊆ (𝐴 +𝑐 1𝑜))
10 0lt1o 7739 . . . . . . . . . 10 ∅ ∈ 1𝑜
11 1oex 7722 . . . . . . . . . . 11 1𝑜 ∈ V
1211snid 4348 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
13 opelxpi 5289 . . . . . . . . . 10 ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}))
1410, 12, 13mp2an 666 . . . . . . . . 9 ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})
15 elun2 3933 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1614, 15mp1i 13 . . . . . . . 8 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1716, 8eleqtrrd 2853 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
18 1n0 7730 . . . . . . . 8 1𝑜 ≠ ∅
19 opelxp2 5292 . . . . . . . . . 10 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 ∈ {∅})
20 elsni 4334 . . . . . . . . . 10 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
2119, 20syl 17 . . . . . . . . 9 (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 = ∅)
2221necon3ai 2968 . . . . . . . 8 (1𝑜 ≠ ∅ → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
2318, 22mp1i 13 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
249, 17, 23ssnelpssd 3870 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜))
25 0ex 4925 . . . . . . . 8 ∅ ∈ V
26 xpsneng 8202 . . . . . . . 8 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
276, 25, 26sylancl 568 . . . . . . 7 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
28 entr 8162 . . . . . . 7 (((𝐴 × {∅}) ≈ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜)) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
2927, 28mpancom 662 . . . . . 6 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
30 fin4i 9323 . . . . . 6 (((𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜) ∧ (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜)) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
3124, 29, 30syl2anc 567 . . . . 5 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
32 fin4en1 9334 . . . . 5 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV → (𝐴 +𝑐 1𝑜) ∈ FinIV))
3331, 32mtod 189 . . . 4 (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ∈ FinIV)
3433con2i 136 . . 3 (𝐴 ∈ FinIV → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
35 brsdom 8133 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) ↔ (𝐴 ≼ (𝐴 +𝑐 1𝑜) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜)))
363, 34, 35sylanbrc 566 . 2 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))
37 sdomnen 8139 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
38 infcda1 9218 . . . . 5 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
3938ensymd 8161 . . . 4 (ω ≼ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜))
4037, 39nsyl 137 . . 3 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ ω ≼ 𝐴)
41 relsdom 8117 . . . . 5 Rel ≺
4241brrelexi 5299 . . . 4 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
43 isfin4-2 9339 . . . 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 1631  wcel 2145  wne 2943  Vcvv 3351  cun 3722  wpss 3725  c0 4064  {csn 4317  cop 4323   class class class wbr 4787   × cxp 5248  Oncon0 5867  (class class class)co 6794  ωcom 7213  1𝑜c1o 7707  cen 8107  cdom 8108  csdm 8109   +𝑐 ccda 9192  FinIVcfin4 9305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-er 7897  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-cda 9193  df-fin4 9312
This theorem is referenced by:  fin45  9417  finngch  9680  gchinf  9682
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