Theorem infcda1 9015

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 7961	. . . . . . . 8 ⊢ Rel ≼
2	1	brrelex2i 5159	. . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
3		1on 7567	. . . . . . 7 ⊢ 1𝑜 ∈ On
4		cdaval 8992	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
5	2, 3, 4	sylancl 694	. . . . . 6 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
6		df1o2 7572	. . . . . . . . 9 ⊢ 1𝑜 = {∅}
7	6	xpeq1i 5135	. . . . . . . 8 ⊢ (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
8		0ex 4790	. . . . . . . . 9 ⊢ ∅ ∈ V
9	3	elexi 3213	. . . . . . . . 9 ⊢ 1𝑜 ∈ V
10	8, 9	xpsn 6407	. . . . . . . 8 ⊢ ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
11	7, 10	eqtr2i 2645	. . . . . . 7 ⊢ {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜})
12	11	a1i 11	. . . . . 6 ⊢ (ω ≼ 𝐴 → {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜}))
13	5, 12	difeq12d 3729	. . . . 5 ⊢ (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})))
14		difun2 4048	. . . . . 6 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
15		xp01disj 7576	. . . . . . 7 ⊢ ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
16		disj3 4021	. . . . . . 7 ⊢ (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
17	15, 16	mpbi 220	. . . . . 6 ⊢ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
18	14, 17	eqtr4i 2647	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (𝐴 × {∅})
19	13, 18	syl6eq 2672	. . . 4 ⊢ (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
20		cdadom3 9010	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
21	2, 3, 20	sylancl 694	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
22		domtr 8009	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ (𝐴 +𝑐 1𝑜)) → ω ≼ (𝐴 +𝑐 1𝑜))
23	21, 22	mpdan 702	. . . . 5 ⊢ (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 1𝑜))
24		infdifsn 8554	. . . . 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 4677	. . 3 ⊢ (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
27	26	ensymd 8007	. 2 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}))
28		xpsneng 8045	. . 3 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
29	2, 8, 28	sylancl 694	. 2 ⊢ (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
30		entr 8008	. 2 ⊢ (((𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
31	27, 29, 30	syl2anc 693	1 ⊢ (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573  ∅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   +_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-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-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-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-cda 8990

This theorem is referenced by:  pwcdaidm  9017  isfin4-3  9137  canthp1lem2  9475