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 ∈ FinIV ↔ 𝐴 ≺ (𝐴 +𝑐 1𝑜))

Proof of Theorem isfin4-3

Step	Hyp	Ref	Expression
1		1on 7454	. . . 4 ⊢ 1𝑜 ∈ On
2		cdadom3 8893	. . . 4 ⊢ ((𝐴 ∈ FinIV ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
3	1, 2	mpan2 703	. . 3 ⊢ (𝐴 ∈ FinIV → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
4		ssun1 3738	. . . . . . . 8 ⊢ (𝐴 × {∅}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
5		relen 7846	. . . . . . . . . 10 ⊢ Rel ≈
6	5	brrelexi 5082	. . . . . . . . 9 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
7		cdaval 8875	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
8	6, 1, 7	sylancl 693	. . . . . . . 8 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
9	4, 8	syl5sseqr 3617	. . . . . . 7 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊆ (𝐴 +𝑐 1𝑜))
10		0lt1o 7471	. . . . . . . . . 10 ⊢ ∅ ∈ 1𝑜
11	1	elexi 3186	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ V
12	11	snid 4155	. . . . . . . . . 10 ⊢ 1𝑜 ∈ {1𝑜}
13		opelxpi 5072	. . . . . . . . . 10 ⊢ ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ {1𝑜}) → ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}))
14	10, 12, 13	mp2an 704	. . . . . . . . 9 ⊢ ⟨∅, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})
15		elun2 3743	. . . . . . . . 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 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 = ∅)
21	19, 20	syl 17	. . . . . . . . 9 ⊢ (⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 = ∅)
22	21	necon3ai 2807	. . . . . . . 8 ⊢ (1𝑜 ≠ ∅ → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
23	18, 22	mp1i 13	. . . . . . 7 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ ⟨∅, 1𝑜⟩ ∈ (𝐴 × {∅}))
24	9, 17, 23	ssnelpssd 3681	. . . . . 6 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ⊊ (𝐴 +𝑐 1𝑜))
25		0ex 4718	. . . . . . . 8 ⊢ ∅ ∈ V
26		xpsneng 7930	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
27	6, 25, 26	sylancl 693	. . . . . . 7 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
28		entr 7894	. . . . . . 7 ⊢ (((𝐴 × {∅}) ≈ 𝐴 ∧ 𝐴 ≈ (𝐴 +𝑐 1𝑜)) → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
29	27, 28	mpancom 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)
31	24, 29, 30	syl2anc 691	. . . . 5 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ (𝐴 +𝑐 1𝑜) ∈ FinIV)
32		fin4en1 9014	. . . . 5 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ FinIV → (𝐴 +𝑐 1𝑜) ∈ FinIV))
33	31, 32	mtod 188	. . . 4 ⊢ (𝐴 ≈ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ∈ FinIV)
34	33	con2i 133	. . 3 ⊢ (𝐴 ∈ FinIV → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
35		brsdom 7864	. . 3 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) ↔ (𝐴 ≼ (𝐴 +𝑐 1𝑜) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜)))
36	3, 34, 35	sylanbrc 695	. 2 ⊢ (𝐴 ∈ FinIV → 𝐴 ≺ (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)
39	38	ensymd 7893	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 +𝑐 1𝑜))
40	37, 39	nsyl 134	. . 3 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ ω ≼ 𝐴)
41		relsdom 7848	. . . . 5 ⊢ Rel ≺
42	41	brrelexi 5082	. . . 4 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
43		isfin4-2 9019	. . . 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 246	. 2 ⊢ (𝐴 ≺ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ FinIV)
46	36, 45	impbii 198	1 ⊢ (𝐴 ∈ FinIV ↔ 𝐴 ≺ (𝐴 +𝑐 1𝑜))

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  1_𝑜c1o 7440   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840   +_𝑐 ccda 8872  Fin^IVcfin4 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