Theorem cda1dif 8998

Description: Adding and subtracting one gives back the original set. Similar to pncan 10287 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
cda1dif	⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)

Proof of Theorem cda1dif

Step	Hyp	Ref	Expression
1		ovexd 6680	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) ∈ V)
2		id 22	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐵 ∈ (𝐴 +𝑐 1𝑜))
3		df1o2 7572	. . . . . . . 8 ⊢ 1𝑜 = {∅}
4	3	xpeq1i 5135	. . . . . . 7 ⊢ (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
5		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
6		1on 7567	. . . . . . . . 9 ⊢ 1𝑜 ∈ On
7	6	elexi 3213	. . . . . . . 8 ⊢ 1𝑜 ∈ V
8	5, 7	xpsn 6407	. . . . . . 7 ⊢ ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
9	4, 8	eqtri 2644	. . . . . 6 ⊢ (1𝑜 × {1𝑜}) = {⟨∅, 1𝑜⟩}
10		ssun2 3777	. . . . . 6 ⊢ (1𝑜 × {1𝑜}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
11	9, 10	eqsstr3i 3636	. . . . 5 ⊢ {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
12		opex 4932	. . . . . 6 ⊢ ⟨∅, 1𝑜⟩ ∈ V
13	12	snss 4316	. . . . 5 ⊢ (⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ↔ {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
14	11, 13	mpbir 221	. . . 4 ⊢ ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
15		relxp 5227	. . . . . . . 8 ⊢ Rel (V × V)
16		cdafn 8991	. . . . . . . . . 10 ⊢ +𝑐 Fn (V × V)
17		fndm 5990	. . . . . . . . . 10 ⊢ ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
18	16, 17	ax-mp 5	. . . . . . . . 9 ⊢ dom +𝑐 = (V × V)
19	18	releqi 5202	. . . . . . . 8 ⊢ (Rel dom +𝑐 ↔ Rel (V × V))
20	15, 19	mpbir 221	. . . . . . 7 ⊢ Rel dom +𝑐
21	20	ovrcl 6686	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ V ∧ 1𝑜 ∈ V))
22	21	simpld 475	. . . . 5 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
23		cdaval 8992	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
24	22, 6, 23	sylancl 694	. . . 4 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
25	14, 24	syl5eleqr 2708	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
26		difsnen 8042	. . 3 ⊢ (((𝐴 +𝑐 1𝑜) ∈ V ∧ 𝐵 ∈ (𝐴 +𝑐 1𝑜) ∧ ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜)) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
27	1, 2, 25, 26	syl3anc 1326	. 2 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
28	24	difeq1d 3727	. . . 4 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩}))
29		xp01disj 7576	. . . . . 6 ⊢ ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
30		disj3 4021	. . . . . 6 ⊢ (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
31	29, 30	mpbi 220	. . . . 5 ⊢ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
32		difun2 4048	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
33	9	difeq2i 3725	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
34	31, 32, 33	3eqtr2i 2650	. . . 4 ⊢ (𝐴 × {∅}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
35	28, 34	syl6eqr 2674	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
36		xpsneng 8045	. . . 4 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
37	22, 5, 36	sylancl 694	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
38	35, 37	eqbrtrd 4675	. 2 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴)
39		entr 8008	. 2 ⊢ ((((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ∧ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
40	27, 38, 39	syl2anc 693	1 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653   × cxp 5112  dom cdm 5114  Rel wrel 5119  Oncon0 5723   Fn wfn 5883  (class class class)co 6650  1_𝑜c1o 7553   ≈ cen 7952   +_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-ord 5726  df-on 5727  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-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-cda 8990

This theorem is referenced by:  canthp1  9476