Theorem lemsuccf 35983

Description: Lemma for unfolding different forms 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
lemsuccf	⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem lemsuccf

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

Step	Hyp	Ref	Expression
1		opex 5402	. . . 4 ⊢ ⟨𝐴, {𝐴}⟩ ∈ V
2		breq1 5092	. . . 4 ⊢ (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
3	1, 2	ceqsexv 3486	. . 3 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
4		brsuccf.1	. . . 4 ⊢ 𝐴 ∈ V
5		snex 5372	. . . 4 ⊢ {𝐴} ∈ V
6		brsuccf.2	. . . 4 ⊢ 𝐵 ∈ V
7	4, 5, 6	brcup 35981	. . 3 ⊢ (⟨𝐴, {𝐴}⟩Cup𝐵 ↔ 𝐵 = (𝐴 ∪ {𝐴}))
8	3, 7	bitri 275	. 2 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
9	4	brtxp2 35923	. . . . 5 ⊢ (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏))
10	9	anbi1i 624	. . . 4 ⊢ ((𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
11		3anass 1094	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
12	11	anbi1i 624	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
13		an32 646	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
14		vex 3440	. . . . . . . . . . . 12 ⊢ 𝑎 ∈ V
15	14	ideq 5791	. . . . . . . . . . 11 ⊢ (𝐴 I 𝑎 ↔ 𝐴 = 𝑎)
16		eqcom 2738	. . . . . . . . . . 11 ⊢ (𝐴 = 𝑎 ↔ 𝑎 = 𝐴)
17	15, 16	bitri 275	. . . . . . . . . 10 ⊢ (𝐴 I 𝑎 ↔ 𝑎 = 𝐴)
18		vex 3440	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
19	4, 18	brsingle 35959	. . . . . . . . . 10 ⊢ (𝐴Singleton𝑏 ↔ 𝑏 = {𝐴})
20	17, 19	anbi12i 628	. . . . . . . . 9 ⊢ ((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴}))
21	20	anbi1i 624	. . . . . . . 8 ⊢ (((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
22		ancom 460	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ↔ ((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
23		df-3an 1088	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ ((𝑎 = 𝐴 ∧ 𝑏 = {𝐴}) ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
24	21, 22, 23	3bitr4i 303	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
25	12, 13, 24	3bitri 297	. . . . . 6 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
26	25	2exbii 1850	. . . . 5 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
27		19.41vv 1951	. . . . 5 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
28		opeq1 4822	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
29	28	eqeq2d 2742	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
30	29	anbi1d 631	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
31		opeq2 4823	. . . . . . . 8 ⊢ (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
32	31	eqeq2d 2742	. . . . . . 7 ⊢ (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
33	32	anbi1d 631	. . . . . 6 ⊢ (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
34	4, 5, 30, 33	ceqsex2v 3490	. . . . 5 ⊢ (∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
35	26, 27, 34	3bitr3i 301	. . . 4 ⊢ ((∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
36	10, 35	bitri 275	. . 3 ⊢ ((𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
37	36	exbii 1849	. 2 ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
38		df-suc 6312	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
39	38	eqeq2i 2744	. 2 ⊢ (𝐵 = suc 𝐴 ↔ 𝐵 = (𝐴 ∪ {𝐴}))
40	8, 37, 39	3bitr4i 303	1 ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436   ∪ cun 3895  {csn 4573  ⟨cop 4579   class class class wbr 5089   I cid 5508  suc csuc 6308   ⊗ ctxp 35872  Singletoncsingle 35880  Cupccup 35888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-symdif 4200  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-eprel 5514  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fo 6487  df-fv 6489  df-1st 7921  df-2nd 7922  df-txp 35896  df-singleton 35904  df-cup 35911

This theorem is referenced by:  brsuccf  35984  dfsuccf2  35985