Theorem brsuccf 32048

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 31979	. . 3 ⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
2	1	breqi 4659	. 2 ⊢ (𝐴Succ𝐵 ↔ 𝐴(Cup ∘ ( I ⊗ Singleton))𝐵)
3		brsuccf.1	. . 3 ⊢ 𝐴 ∈ V
4		brsuccf.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brco 5292	. 2 ⊢ (𝐴(Cup ∘ ( I ⊗ Singleton))𝐵 ↔ ∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵))
6		opex 4932	. . . . 5 ⊢ ⟨𝐴, {𝐴}⟩ ∈ V
7		breq1 4656	. . . . 5 ⊢ (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
8	6, 7	ceqsexv 3242	. . . 4 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
9		snex 4908	. . . . 5 ⊢ {𝐴} ∈ V
10	3, 9, 4	brcup 32046	. . . 4 ⊢ (⟨𝐴, {𝐴}⟩Cup𝐵 ↔ 𝐵 = (𝐴 ∪ {𝐴}))
11	8, 10	bitri 264	. . 3 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
12	3	brtxp2 31988	. . . . . 6 ⊢ (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏))
13	12	anbi1i 731	. . . . 5 ⊢ ((𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
14		3anass 1042	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
15	14	anbi1i 731	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
16		an32 839	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
17		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
18	17	ideq 5274	. . . . . . . . . . . 12 ⊢ (𝐴 I 𝑎 ↔ 𝐴 = 𝑎)
19		eqcom 2629	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝑎 ↔ 𝑎 = 𝐴)
20	18, 19	bitri 264	. . . . . . . . . . 11 ⊢ (𝐴 I 𝑎 ↔ 𝑎 = 𝐴)
21		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑏 ∈ V
22	3, 21	brsingle 32024	. . . . . . . . . . 11 ⊢ (𝐴Singleton𝑏 ↔ 𝑏 = {𝐴})
23	20, 22	anbi12i 733	. . . . . . . . . 10 ⊢ ((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴}))
24	23	anbi1i 731	. . . . . . . . 9 ⊢ (((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
25		ancom 466	. . . . . . . . 9 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ↔ ((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
26		df-3an 1039	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
27	24, 25, 26	3bitr4i 292	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
28	15, 16, 27	3bitri 286	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
29	28	2exbii 1775	. . . . . 6 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
30		19.41vv 1915	. . . . . 6 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
31		opeq1 4402	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
32	31	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
33	32	anbi1d 741	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
34		opeq2 4403	. . . . . . . . 9 ⊢ (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
35	34	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
36	35	anbi1d 741	. . . . . . 7 ⊢ (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
37	3, 9, 33, 36	ceqsex2v 3245	. . . . . 6 ⊢ (∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
38	29, 30, 37	3bitr3i 290	. . . . 5 ⊢ ((∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
39	13, 38	bitri 264	. . . 4 ⊢ ((𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
40	39	exbii 1774	. . 3 ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
41		df-suc 5729	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
42	41	eqeq2i 2634	. . 3 ⊢ (𝐵 = suc 𝐴 ↔ 𝐵 = (𝐴 ∪ {𝐴}))
43	11, 40, 42	3bitr4i 292	. 2 ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)
44	2, 5, 43	3bitri 286	1 ⊢ (𝐴Succ𝐵 ↔ 𝐵 = suc 𝐴)

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572  {csn 4177  ⟨cop 4183   class class class wbr 4653   I cid 5023   ∘ ccom 5118  suc csuc 5725   ⊗ ctxp 31937  Singletoncsingle 31945  Cupccup 31953  Succcsuccf 31955

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-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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-singleton 31969  df-cup 31976  df-succf 31979

This theorem is referenced by:  dfrdg4  32058