Theorem infcda1 9053

Description: An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
infcda1	⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)

Proof of Theorem infcda1

Step	Hyp	Ref	Expression
1		reldom 8003	. . . . . . . 8 ⊢ Rel ≼
2	1	brrelex2i 5193	. . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
3		1on 7612	. . . . . . 7 ⊢ 1𝑜 ∈ On
4		cdaval 9030	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
5	2, 3, 4	sylancl 695	. . . . . 6 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
6		df1o2 7617	. . . . . . . . 9 ⊢ 1𝑜 = {∅}
7	6	xpeq1i 5169	. . . . . . . 8 ⊢ (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
8		0ex 4823	. . . . . . . . 9 ⊢ ∅ ∈ V
9	3	elexi 3244	. . . . . . . . 9 ⊢ 1𝑜 ∈ V
10	8, 9	xpsn 6447	. . . . . . . 8 ⊢ ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
11	7, 10	eqtr2i 2674	. . . . . . 7 ⊢ {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜})
12	11	a1i 11	. . . . . 6 ⊢ (ω ≼ 𝐴 → {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜}))
13	5, 12	difeq12d 3762	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})))
14		difun2 4081	. . . . . 6 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
15		xp01disj 7621	. . . . . . 7 ⊢ ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
16		disj3 4054	. . . . . . 7 ⊢ (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
17	15, 16	mpbi 220	. . . . . 6 ⊢ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
18	14, 17	eqtr4i 2676	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (𝐴 × {∅})
19	13, 18	syl6eq 2701	. . . 4 ⊢ (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
20		cdadom3 9048	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
21	2, 3, 20	sylancl 695	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
22		domtr 8050	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 +𝑐 1𝑜)) → ω ≼ (𝐴 +𝑐 1𝑜))
23	21, 22	mpdan 703	. . . . 5 ⊢ (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 1𝑜))
24		infdifsn 8592	. . . . 5 ⊢ (ω ≼ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ (𝐴 +𝑐 1𝑜))
25	23, 24	syl 17	. . . 4 ⊢ (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ (𝐴 +𝑐 1𝑜))
26	19, 25	eqbrtrrd 4709	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
27	26	ensymd 8048	. 2 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}))
28		xpsneng 8086	. . 3 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
29	2, 8, 28	sylancl 695	. 2 ⊢ (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
30		entr 8049	. 2 ⊢ (((𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
31	27, 29, 30	syl2anc 694	1 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606  ∅c0 3948  {csn 4210  ⟨cop 4216   class class class wbr 4685   × cxp 5141  Oncon0 5761  (class class class)co 6690  ωcom 7107  1_𝑜c1o 7598   ≈ cen 7994   ≼ cdom 7995   +_𝑐 ccda 9027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-cda 9028

This theorem is referenced by:  pwcdaidm  9055  isfin4-3  9175  canthp1lem2  9513