Theorem cda1dif 8881

Description: Adding and subtracting one gives back the original set. Similar to pncan 10166 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		ovex 6577	. . . 4 ⊢ (𝐴 +𝑐 1𝑜) ∈ V
2	1	a1i 11	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) ∈ V)
3		id 22	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐵 ∈ (𝐴 +𝑐 1𝑜))
4		df1o2 7459	. . . . . . . 8 ⊢ 1𝑜 = {∅}
5	4	xpeq1i 5059	. . . . . . 7 ⊢ (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
6		0ex 4718	. . . . . . . 8 ⊢ ∅ ∈ V
7		1on 7454	. . . . . . . . 9 ⊢ 1𝑜 ∈ On
8	7	elexi 3186	. . . . . . . 8 ⊢ 1𝑜 ∈ V
9	6, 8	xpsn 6313	. . . . . . 7 ⊢ ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
10	5, 9	eqtri 2632	. . . . . 6 ⊢ (1𝑜 × {1𝑜}) = {⟨∅, 1𝑜⟩}
11		ssun2 3739	. . . . . 6 ⊢ (1𝑜 × {1𝑜}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
12	10, 11	eqsstr3i 3599	. . . . 5 ⊢ {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
13		opex 4859	. . . . . 6 ⊢ ⟨∅, 1𝑜⟩ ∈ V
14	13	snss 4259	. . . . 5 ⊢ (⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ↔ {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
15	12, 14	mpbir 220	. . . 4 ⊢ ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
16		relxp 5150	. . . . . . . 8 ⊢ Rel (V × V)
17		cdafn 8874	. . . . . . . . . 10 ⊢ +𝑐 Fn (V × V)
18		fndm 5904	. . . . . . . . . 10 ⊢ ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
19	17, 18	ax-mp 5	. . . . . . . . 9 ⊢ dom +𝑐 = (V × V)
20	19	releqi 5125	. . . . . . . 8 ⊢ (Rel dom +𝑐 ↔ Rel (V × V))
21	16, 20	mpbir 220	. . . . . . 7 ⊢ Rel dom +𝑐
22	21	ovrcl 6584	. . . . . 6 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ V ∧ 1𝑜 ∈ V))
23	22	simpld 474	. . . . 5 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
24		cdaval 8875	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
25	23, 7, 24	sylancl 693	. . . 4 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
26	15, 25	syl5eleqr 2695	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
27		difsnen 7927	. . 3 ⊢ (((𝐴 +𝑐 1𝑜) ∈ V ∧ 𝐵 ∈ (𝐴 +𝑐 1𝑜) ∧ ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜)) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
28	2, 3, 26, 27	syl3anc 1318	. 2 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
29	25	difeq1d 3689	. . . 4 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩}))
30		xp01disj 7463	. . . . . 6 ⊢ ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
31		disj3 3973	. . . . . 6 ⊢ (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
32	30, 31	mpbi 219	. . . . 5 ⊢ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
33		difun2 4000	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
34	10	difeq2i 3687	. . . . 5 ⊢ (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
35	32, 33, 34	3eqtr2i 2638	. . . 4 ⊢ (𝐴 × {∅}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
36	29, 35	syl6eqr 2662	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
37		xpsneng 7930	. . . 4 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
38	23, 6, 37	sylancl 693	. . 3 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
39	36, 38	eqbrtrd 4605	. 2 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴)
40		entr 7894	. 2 ⊢ ((((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ∧ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
41	28, 39, 40	syl2anc 691	1 ⊢ (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038  Rel wrel 5043  Oncon0 5640   Fn wfn 5799  (class class class)co 6549  1_𝑜c1o 7440   ≈ cen 7838   +_𝑐 ccda 8872

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-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-ord 5643  df-on 5644  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-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-en 7842  df-cda 8873

This theorem is referenced by:  canthp1  9355