Theorem cdainf 9014

Description: A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
cdainf	⊢ (ω ≼ 𝐴 ↔ ω ≼ (𝐴 +𝑐 𝐴))

Proof of Theorem cdainf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 7961	. . . . 5 ⊢ Rel ≼
2	1	brrelex2i 5159	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
3		cdadom3 9010	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ≼ (𝐴 +𝑐 𝐴))
4	2, 2, 3	syl2anc 693	. . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 +𝑐 𝐴))
5		domtr 8009	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 +𝑐 𝐴)) → ω ≼ (𝐴 +𝑐 𝐴))
6	4, 5	mpdan 702	. 2 ⊢ (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 𝐴))
7		infn0 8222	. . . 4 ⊢ (ω ≼ (𝐴 +𝑐 𝐴) → (𝐴 +𝑐 𝐴) ≠ ∅)
8		cdafn 8991	. . . . . . . 8 ⊢ +𝑐 Fn (V × V)
9		fndm 5990	. . . . . . . 8 ⊢ ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ dom +𝑐 = (V × V)
11	10	ndmov 6818	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ∅)
12	11	necon1ai 2821	. . . . 5 ⊢ ((𝐴 +𝑐 𝐴) ≠ ∅ → (𝐴 ∈ V ∧ 𝐴 ∈ V))
13	12	simpld 475	. . . 4 ⊢ ((𝐴 +𝑐 𝐴) ≠ ∅ → 𝐴 ∈ V)
14	7, 13	syl 17	. . 3 ⊢ (ω ≼ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V)
15		ovex 6678	. . . . 5 ⊢ (𝐴 +𝑐 𝐴) ∈ V
16	15	domen 7968	. . . 4 ⊢ (ω ≼ (𝐴 +𝑐 𝐴) ↔ ∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴)))
17		indi 3873	. . . . . . . . 9 ⊢ (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜})))
18		simprr 796	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ (𝐴 +𝑐 𝐴))
19		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ (ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝐴 ∈ V)
20		cdaval 8992	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
21	19, 19, 20	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
22	18, 21	sseqtrd 3641	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
23		df-ss 3588	. . . . . . . . . 10 ⊢ (𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})) ↔ (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = 𝑥)
24	22, 23	sylib 208	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = 𝑥)
25	17, 24	syl5eqr 2670	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) = 𝑥)
26		ensym 8005	. . . . . . . . 9 ⊢ (ω ≈ 𝑥 → 𝑥 ≈ ω)
27	26	ad2antrl 764	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ≈ ω)
28	25, 27	eqbrtrd 4675	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ (ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω)
29	28	ex 450	. . . . . 6 ⊢ (𝐴 ∈ V → ((ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω))
30		cdainflem 9013	. . . . . . 7 ⊢ (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω ∨ (𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω))
31		snex 4908	. . . . . . . . . . . 12 ⊢ {∅} ∈ V
32		xpexg 6960	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
33	31, 32	mpan2 707	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
34		inss2 3834	. . . . . . . . . . 11 ⊢ (𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅})
35		ssdomg 8001	. . . . . . . . . . 11 ⊢ ((𝐴 × {∅}) ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅}) → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅})))
36	33, 34, 35	mpisyl 21	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}))
37		0ex 4790	. . . . . . . . . . 11 ⊢ ∅ ∈ V
38		xpsneng 8045	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
39	37, 38	mpan2 707	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 × {∅}) ≈ 𝐴)
40		domentr 8015	. . . . . . . . . 10 ⊢ (((𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
41	36, 39, 40	syl2anc 693	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
42		domen1 8102	. . . . . . . . 9 ⊢ ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴 ↔ ω ≼ 𝐴))
43	41, 42	syl5ibcom 235	. . . . . . . 8 ⊢ (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ω ≼ 𝐴))
44		snex 4908	. . . . . . . . . . . 12 ⊢ {1𝑜} ∈ V
45		xpexg 6960	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ {1𝑜} ∈ V) → (𝐴 × {1𝑜}) ∈ V)
46	44, 45	mpan2 707	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 × {1𝑜}) ∈ V)
47		inss2 3834	. . . . . . . . . . 11 ⊢ (𝑥 ∩ (𝐴 × {1𝑜})) ⊆ (𝐴 × {1𝑜})
48		ssdomg 8001	. . . . . . . . . . 11 ⊢ ((𝐴 × {1𝑜}) ∈ V → ((𝑥 ∩ (𝐴 × {1𝑜})) ⊆ (𝐴 × {1𝑜}) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜})))
49	46, 47, 48	mpisyl 21	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜}))
50		1on 7567	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
51		xpsneng 8045	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
52	50, 51	mpan2 707	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
53		domentr 8015	. . . . . . . . . 10 ⊢ (((𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜}) ∧ (𝐴 × {1𝑜}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
54	49, 52, 53	syl2anc 693	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
55		domen1 8102	. . . . . . . . 9 ⊢ ((𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω → ((𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴 ↔ ω ≼ 𝐴))
56	54, 55	syl5ibcom 235	. . . . . . . 8 ⊢ (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω → ω ≼ 𝐴))
57	43, 56	jaod 395	. . . . . . 7 ⊢ (𝐴 ∈ V → (((𝑥 ∩ (𝐴 × {∅})) ≈ ω ∨ (𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω) → ω ≼ 𝐴))
58	30, 57	syl5 34	. . . . . 6 ⊢ (𝐴 ∈ V → (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω → ω ≼ 𝐴))
59	29, 58	syld 47	. . . . 5 ⊢ (𝐴 ∈ V → ((ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ω ≼ 𝐴))
60	59	exlimdv 1861	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ω ≼ 𝐴))
61	16, 60	syl5bi 232	. . 3 ⊢ (𝐴 ∈ V → (ω ≼ (𝐴 +𝑐 𝐴) → ω ≼ 𝐴))
62	14, 61	mpcom 38	. 2 ⊢ (ω ≼ (𝐴 +𝑐 𝐴) → ω ≼ 𝐴)
63	6, 62	impbii 199	1 ⊢ (ω ≼ 𝐴 ↔ ω ≼ (𝐴 +𝑐 𝐴))

Syntax hints:   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  dom cdm 5114  Oncon0 5723   Fn wfn 5883  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553   ≈ cen 7952   ≼ cdom 7953   +_𝑐 ccda 8989

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-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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-cda 8990

This theorem is referenced by:  infdif  9031