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Theorem isfin4p1 9729
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9711 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
isfin4p1 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 ⊔ 1o))

Proof of Theorem isfin4p1
StepHypRef Expression
1 1on 8101 . . . 4 1o ∈ On
2 djudoml 9602 . . . 4 ((𝐴 ∈ FinIV ∧ 1o ∈ On) → 𝐴 ≼ (𝐴 ⊔ 1o))
31, 2mpan2 689 . . 3 (𝐴 ∈ FinIV𝐴 ≼ (𝐴 ⊔ 1o))
4 1oex 8102 . . . . . . . . . . 11 1o ∈ V
54snid 4593 . . . . . . . . . 10 1o ∈ {1o}
6 0lt1o 8121 . . . . . . . . . 10 ∅ ∈ 1o
7 opelxpi 5585 . . . . . . . . . 10 ((1o ∈ {1o} ∧ ∅ ∈ 1o) → ⟨1o, ∅⟩ ∈ ({1o} × 1o))
85, 6, 7mp2an 690 . . . . . . . . 9 ⟨1o, ∅⟩ ∈ ({1o} × 1o)
9 elun2 4151 . . . . . . . . 9 (⟨1o, ∅⟩ ∈ ({1o} × 1o) → ⟨1o, ∅⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 1o)))
108, 9ax-mp 5 . . . . . . . 8 ⟨1o, ∅⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 1o))
11 df-dju 9322 . . . . . . . 8 (𝐴 ⊔ 1o) = (({∅} × 𝐴) ∪ ({1o} × 1o))
1210, 11eleqtrri 2910 . . . . . . 7 ⟨1o, ∅⟩ ∈ (𝐴 ⊔ 1o)
13 1n0 8111 . . . . . . . 8 1o ≠ ∅
14 opelxp1 5589 . . . . . . . . . 10 (⟨1o, ∅⟩ ∈ ({∅} × 𝐴) → 1o ∈ {∅})
15 elsni 4576 . . . . . . . . . 10 (1o ∈ {∅} → 1o = ∅)
1614, 15syl 17 . . . . . . . . 9 (⟨1o, ∅⟩ ∈ ({∅} × 𝐴) → 1o = ∅)
1716necon3ai 3039 . . . . . . . 8 (1o ≠ ∅ → ¬ ⟨1o, ∅⟩ ∈ ({∅} × 𝐴))
1813, 17ax-mp 5 . . . . . . 7 ¬ ⟨1o, ∅⟩ ∈ ({∅} × 𝐴)
19 ssun1 4146 . . . . . . . . 9 ({∅} × 𝐴) ⊆ (({∅} × 𝐴) ∪ ({1o} × 1o))
2019, 11sseqtrri 4002 . . . . . . . 8 ({∅} × 𝐴) ⊆ (𝐴 ⊔ 1o)
21 ssnelpss 4086 . . . . . . . 8 (({∅} × 𝐴) ⊆ (𝐴 ⊔ 1o) → ((⟨1o, ∅⟩ ∈ (𝐴 ⊔ 1o) ∧ ¬ ⟨1o, ∅⟩ ∈ ({∅} × 𝐴)) → ({∅} × 𝐴) ⊊ (𝐴 ⊔ 1o)))
2220, 21ax-mp 5 . . . . . . 7 ((⟨1o, ∅⟩ ∈ (𝐴 ⊔ 1o) ∧ ¬ ⟨1o, ∅⟩ ∈ ({∅} × 𝐴)) → ({∅} × 𝐴) ⊊ (𝐴 ⊔ 1o))
2312, 18, 22mp2an 690 . . . . . 6 ({∅} × 𝐴) ⊊ (𝐴 ⊔ 1o)
24 0ex 5202 . . . . . . . 8 ∅ ∈ V
25 relen 8506 . . . . . . . . 9 Rel ≈
2625brrelex1i 5601 . . . . . . . 8 (𝐴 ≈ (𝐴 ⊔ 1o) → 𝐴 ∈ V)
27 xpsnen2g 8602 . . . . . . . 8 ((∅ ∈ V ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴)
2824, 26, 27sylancr 589 . . . . . . 7 (𝐴 ≈ (𝐴 ⊔ 1o) → ({∅} × 𝐴) ≈ 𝐴)
29 entr 8553 . . . . . . 7 ((({∅} × 𝐴) ≈ 𝐴𝐴 ≈ (𝐴 ⊔ 1o)) → ({∅} × 𝐴) ≈ (𝐴 ⊔ 1o))
3028, 29mpancom 686 . . . . . 6 (𝐴 ≈ (𝐴 ⊔ 1o) → ({∅} × 𝐴) ≈ (𝐴 ⊔ 1o))
31 fin4i 9712 . . . . . 6 ((({∅} × 𝐴) ⊊ (𝐴 ⊔ 1o) ∧ ({∅} × 𝐴) ≈ (𝐴 ⊔ 1o)) → ¬ (𝐴 ⊔ 1o) ∈ FinIV)
3223, 30, 31sylancr 589 . . . . 5 (𝐴 ≈ (𝐴 ⊔ 1o) → ¬ (𝐴 ⊔ 1o) ∈ FinIV)
33 fin4en1 9723 . . . . 5 (𝐴 ≈ (𝐴 ⊔ 1o) → (𝐴 ∈ FinIV → (𝐴 ⊔ 1o) ∈ FinIV))
3432, 33mtod 200 . . . 4 (𝐴 ≈ (𝐴 ⊔ 1o) → ¬ 𝐴 ∈ FinIV)
3534con2i 141 . . 3 (𝐴 ∈ FinIV → ¬ 𝐴 ≈ (𝐴 ⊔ 1o))
36 brsdom 8524 . . 3 (𝐴 ≺ (𝐴 ⊔ 1o) ↔ (𝐴 ≼ (𝐴 ⊔ 1o) ∧ ¬ 𝐴 ≈ (𝐴 ⊔ 1o)))
373, 35, 36sylanbrc 585 . 2 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 ⊔ 1o))
38 sdomnen 8530 . . . 4 (𝐴 ≺ (𝐴 ⊔ 1o) → ¬ 𝐴 ≈ (𝐴 ⊔ 1o))
39 infdju1 9607 . . . . 5 (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ 𝐴)
4039ensymd 8552 . . . 4 (ω ≼ 𝐴𝐴 ≈ (𝐴 ⊔ 1o))
4138, 40nsyl 142 . . 3 (𝐴 ≺ (𝐴 ⊔ 1o) → ¬ ω ≼ 𝐴)
42 relsdom 8508 . . . . 5 Rel ≺
4342brrelex1i 5601 . . . 4 (𝐴 ≺ (𝐴 ⊔ 1o) → 𝐴 ∈ V)
44 isfin4-2 9728 . . . 4 (𝐴 ∈ V → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
4543, 44syl 17 . . 3 (𝐴 ≺ (𝐴 ⊔ 1o) → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
4641, 45mpbird 259 . 2 (𝐴 ≺ (𝐴 ⊔ 1o) → 𝐴 ∈ FinIV)
4737, 46impbii 211 1 (𝐴 ∈ FinIV𝐴 ≺ (𝐴 ⊔ 1o))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1531  wcel 2108  wne 3014  Vcvv 3493  cun 3932  wss 3934  wpss 3935  c0 4289  {csn 4559  cop 4565   class class class wbr 5057   × cxp 5546  Oncon0 6184  ωcom 7572  1oc1o 8087  cen 8498  cdom 8499  csdm 8500  cdju 9319  FinIVcfin4 9694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-fin4 9701
This theorem is referenced by:  fin45  9806  finngch  10069  gchinf  10071
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