Theorem brsuccf 35925

Description: Binary relation form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brsuccf.1	⊢ 𝐴 ∈ V
brsuccf.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
brsuccf	⊢ (𝐴Succ𝐵 ↔ 𝐵 = suc 𝐴)

Proof of Theorem brsuccf

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-succf 35856	. . 3 ⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
2	1	breqi 5098	. 2 ⊢ (𝐴Succ𝐵 ↔ 𝐴(Cup ∘ ( I ⊗ Singleton))𝐵)
3		brsuccf.1	. . 3 ⊢ 𝐴 ∈ V
4		brsuccf.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brco 5813	. 2 ⊢ (𝐴(Cup ∘ ( I ⊗ Singleton))𝐵 ↔ ∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵))
6		opex 5407	. . . . 5 ⊢ ⟨𝐴, {𝐴}⟩ ∈ V
7		breq1 5095	. . . . 5 ⊢ (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
8	6, 7	ceqsexv 3487	. . . 4 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
9		snex 5375	. . . . 5 ⊢ {𝐴} ∈ V
10	3, 9, 4	brcup 35923	. . . 4 ⊢ (⟨𝐴, {𝐴}⟩Cup𝐵 ↔ 𝐵 = (𝐴 ∪ {𝐴}))
11	8, 10	bitri 275	. . 3 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
12	3	brtxp2 35865	. . . . . 6 ⊢ (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏))
13	12	anbi1i 624	. . . . 5 ⊢ ((𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
14		3anass 1094	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
15	14	anbi1i 624	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
16		an32 646	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
17		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
18	17	ideq 5795	. . . . . . . . . . . 12 ⊢ (𝐴 I 𝑎 ↔ 𝐴 = 𝑎)
19		eqcom 2736	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝑎 ↔ 𝑎 = 𝐴)
20	18, 19	bitri 275	. . . . . . . . . . 11 ⊢ (𝐴 I 𝑎 ↔ 𝑎 = 𝐴)
21		vex 3440	. . . . . . . . . . . 12 ⊢ 𝑏 ∈ V
22	3, 21	brsingle 35901	. . . . . . . . . . 11 ⊢ (𝐴Singleton𝑏 ↔ 𝑏 = {𝐴})
23	20, 22	anbi12i 628	. . . . . . . . . 10 ⊢ ((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴}))
24	23	anbi1i 624	. . . . . . . . 9 ⊢ (((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
25		ancom 460	. . . . . . . . 9 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ↔ ((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
26		df-3an 1088	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
27	24, 25, 26	3bitr4i 303	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
28	15, 16, 27	3bitri 297	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
29	28	2exbii 1849	. . . . . 6 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
30		19.41vv 1950	. . . . . 6 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
31		opeq1 4824	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
32	31	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
33	32	anbi1d 631	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
34		opeq2 4825	. . . . . . . . 9 ⊢ (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
35	34	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
36	35	anbi1d 631	. . . . . . 7 ⊢ (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
37	3, 9, 33, 36	ceqsex2v 3491	. . . . . 6 ⊢ (∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
38	29, 30, 37	3bitr3i 301	. . . . 5 ⊢ ((∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
39	13, 38	bitri 275	. . . 4 ⊢ ((𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
40	39	exbii 1848	. . 3 ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
41		df-suc 6313	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
42	41	eqeq2i 2742	. . 3 ⊢ (𝐵 = suc 𝐴 ↔ 𝐵 = (𝐴 ∪ {𝐴}))
43	11, 40, 42	3bitr4i 303	. 2 ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)
44	2, 5, 43	3bitri 297	1 ⊢ (𝐴Succ𝐵 ↔ 𝐵 = suc 𝐴)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3436   ∪ cun 3901  {csn 4577  ⟨cop 4583   class class class wbr 5092   I cid 5513   ∘ ccom 5623  suc csuc 6309   ⊗ ctxp 35814  Singletoncsingle 35822  Cupccup 35830  Succcsuccf 35832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-symdif 4204  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488  df-fv 6490  df-1st 7924  df-2nd 7925  df-txp 35838  df-singleton 35846  df-cup 35853  df-succf 35856

This theorem is referenced by:  dfrdg4  35935