Theorem isfin4-3 9137

Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9119 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 ∈ FinIV ↔ 𝐴 ≺ (𝐴 +𝑐 1𝑜))

Proof of Theorem isfin4-3

Step	Hyp	Ref	Expression
1		1on 7567	. . . 4 ⊢ 1𝑜 ∈ On
2		cdadom3 9010	. . . 4 ⊢ ((𝐴 ∈ FinIV ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
3	1, 2	mpan2 707	. . 3 ⊢ (𝐴 ∈ FinIV → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
4		ssun1 3776	. . . . . . . 8 ⊢ (𝐴 × {∅}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
5		relen 7960	. . . . . . . . . 10 ⊢ Rel ≈
6	5	brrelexi 5158	. . . . . . . . 9 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
7		cdaval 8992	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
8	6, 1, 7	sylancl 694	. . . . . . . 8 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
9	4, 8	syl5sseqr 3654	. . . . . . 7 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊆ (𝐴 +𝑐 1𝑜))
10		0lt1o 7584	. . . . . . . . . 10 ⊢ ∅ ∈ 1𝑜
11	1	elexi 3213	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ V
12	11	snid 4208	. . . . . . . . . 10 ⊢ 1𝑜 ∈ {1𝑜}
13		opelxpi 5148	. . . . . . . . . 10 ⊢ ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}))
14	10, 12, 13	mp2an 708	. . . . . . . . 9 ⊢ ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})
15		elun2 3781	. . . . . . . . 9 ⊢ (⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
16	14, 15	mp1i 13	. . . . . . . 8 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
17	16, 8	eleqtrrd 2704	. . . . . . 7 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
18		1n0 7575	. . . . . . . 8 ⊢ 1𝑜 ≠ ∅
19		opelxp2 5151	. . . . . . . . . 10 ⊢ (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 ∈ {∅})
20		elsni 4194	. . . . . . . . . 10 ⊢ (1𝑜 ∈ {∅} → 1𝑜 = ∅)
21	19, 20	syl 17	. . . . . . . . 9 ⊢ (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 = ∅)
22	21	necon3ai 2819	. . . . . . . 8 ⊢ (1𝑜 ≠ ∅ → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
23	18, 22	mp1i 13	. . . . . . 7 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
24	9, 17, 23	ssnelpssd 3719	. . . . . 6 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜))
25		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
26		xpsneng 8045	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
27	6, 25, 26	sylancl 694	. . . . . . 7 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
28		entr 8008	. . . . . . 7 ⊢ (((𝐴 × {∅}) ≈ 𝐴 ∧ 𝐴 ≈ (𝐴 +𝑐 1𝑜)) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
29	27, 28	mpancom 703	. . . . . 6 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
30		fin4i 9120	. . . . . 6 ⊢ (((𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜) ∧ (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜)) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
31	24, 29, 30	syl2anc 693	. . . . 5 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
32		fin4en1 9131	. . . . 5 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV → (𝐴 +𝑐 1𝑜) ∈ FinIV))
33	31, 32	mtod 189	. . . 4 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ∈ FinIV)
34	33	con2i 134	. . 3 ⊢ (𝐴 ∈ FinIV → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
35		brsdom 7978	. . 3 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) ↔ (𝐴 ≼ (𝐴 +𝑐 1𝑜) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜)))
36	3, 34, 35	sylanbrc 698	. 2 ⊢ (𝐴 ∈ FinIV → 𝐴 ≺ (𝐴 +𝑐 1𝑜))
37		sdomnen 7984	. . . 4 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
38		infcda1 9015	. . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
39	38	ensymd 8007	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 +𝑐 1𝑜))
40	37, 39	nsyl 135	. . 3 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ ω ≼ 𝐴)
41		relsdom 7962	. . . . 5 ⊢ Rel ≺
42	41	brrelexi 5158	. . . 4 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
43		isfin4-2 9136	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
44	42, 43	syl 17	. . 3 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
45	40, 44	mpbird 247	. 2 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ FinIV)
46	36, 45	impbii 199	1 ⊢ (𝐴 ∈ FinIV ↔ 𝐴 ≺ (𝐴 +𝑐 1𝑜))

Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∪ cun 3572   ⊊ wpss 3575  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653   × cxp 5112  Oncon0 5723  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954   +_𝑐 ccda 8989  Fin^IVcfin4 9102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-cda 8990  df-fin4 9109

This theorem is referenced by:  fin45  9214  finngch  9477  gchinf  9479