Theorem brsuccf 33515

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 33446	. . 3 ⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
2	1	breqi 5036	. 2 ⊢ (𝐴Succ𝐵 ↔ 𝐴(Cup ∘ ( I ⊗ Singleton))𝐵)
3		brsuccf.1	. . 3 ⊢ 𝐴 ∈ V
4		brsuccf.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brco 5705	. 2 ⊢ (𝐴(Cup ∘ ( I ⊗ Singleton))𝐵 ↔ ∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵))
6		opex 5321	. . . . 5 ⊢ ⟨𝐴, {𝐴}⟩ ∈ V
7		breq1 5033	. . . . 5 ⊢ (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
8	6, 7	ceqsexv 3489	. . . 4 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
9		snex 5297	. . . . 5 ⊢ {𝐴} ∈ V
10	3, 9, 4	brcup 33513	. . . 4 ⊢ (⟨𝐴, {𝐴}⟩Cup𝐵 ↔ 𝐵 = (𝐴 ∪ {𝐴}))
11	8, 10	bitri 278	. . 3 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
12	3	brtxp2 33455	. . . . . 6 ⊢ (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏))
13	12	anbi1i 626	. . . . 5 ⊢ ((𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
14		3anass 1092	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
15	14	anbi1i 626	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
16		an32 645	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
17		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
18	17	ideq 5687	. . . . . . . . . . . 12 ⊢ (𝐴 I 𝑎 ↔ 𝐴 = 𝑎)
19		eqcom 2805	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝑎 ↔ 𝑎 = 𝐴)
20	18, 19	bitri 278	. . . . . . . . . . 11 ⊢ (𝐴 I 𝑎 ↔ 𝑎 = 𝐴)
21		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑏 ∈ V
22	3, 21	brsingle 33491	. . . . . . . . . . 11 ⊢ (𝐴Singleton𝑏 ↔ 𝑏 = {𝐴})
23	20, 22	anbi12i 629	. . . . . . . . . 10 ⊢ ((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴}))
24	23	anbi1i 626	. . . . . . . . 9 ⊢ (((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
25		ancom 464	. . . . . . . . 9 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ↔ ((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
26		df-3an 1086	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
27	24, 25, 26	3bitr4i 306	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
28	15, 16, 27	3bitri 300	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
29	28	2exbii 1850	. . . . . 6 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
30		19.41vv 1951	. . . . . 6 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
31		opeq1 4763	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
32	31	eqeq2d 2809	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
33	32	anbi1d 632	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
34		opeq2 4765	. . . . . . . . 9 ⊢ (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
35	34	eqeq2d 2809	. . . . . . . 8 ⊢ (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
36	35	anbi1d 632	. . . . . . 7 ⊢ (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
37	3, 9, 33, 36	ceqsex2v 3492	. . . . . 6 ⊢ (∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
38	29, 30, 37	3bitr3i 304	. . . . 5 ⊢ ((∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
39	13, 38	bitri 278	. . . 4 ⊢ ((𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
40	39	exbii 1849	. . 3 ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
41		df-suc 6165	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
42	41	eqeq2i 2811	. . 3 ⊢ (𝐵 = suc 𝐴 ↔ 𝐵 = (𝐴 ∪ {𝐴}))
43	11, 40, 42	3bitr4i 306	. 2 ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)
44	2, 5, 43	3bitri 300	1 ⊢ (𝐴Succ𝐵 ↔ 𝐵 = suc 𝐴)

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879  {csn 4525  ⟨cop 4531   class class class wbr 5030   I cid 5424   ∘ ccom 5523  suc csuc 6161   ⊗ ctxp 33404  Singletoncsingle 33412  Cupccup 33420  Succcsuccf 33422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-symdif 4169  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-eprel 5430  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-1st 7671  df-2nd 7672  df-txp 33428  df-singleton 33436  df-cup 33443  df-succf 33446

This theorem is referenced by:  dfrdg4  33525