Theorem lemsuccf 36155

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 5419	. . . 4 ⊢ ⟨𝐴, {𝐴}⟩ ∈ V
2		breq1 5103	. . . 4 ⊢ (𝑥 = ⟨𝐴, {𝐴}⟩ → (𝑥Cup𝐵 ↔ ⟨𝐴, {𝐴}⟩Cup𝐵))
3	1, 2	ceqsexv 3492	. . 3 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ ⟨𝐴, {𝐴}⟩Cup𝐵)
4		brsuccf.1	. . . 4 ⊢ 𝐴 ∈ V
5		snex 5385	. . . 4 ⊢ {𝐴} ∈ V
6		brsuccf.2	. . . 4 ⊢ 𝐵 ∈ V
7	4, 5, 6	brcup 36153	. . 3 ⊢ (⟨𝐴, {𝐴}⟩Cup𝐵 ↔ 𝐵 = (𝐴 ∪ {𝐴}))
8	3, 7	bitri 275	. 2 ⊢ (∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵) ↔ 𝐵 = (𝐴 ∪ {𝐴}))
9	4	brtxp2 36095	. . . . 5 ⊢ (𝐴( I ⊗ Singleton)𝑥 ↔ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏))
10	9	anbi1i 625	. . . 4 ⊢ ((𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
11		3anass 1095	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
12	11	anbi1i 625	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵))
13		an32 647	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)) ∧ 𝑥Cup𝐵) ↔ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ∧ (𝐴 I 𝑎 ∧ 𝐴Singleton𝑏)))
14		vex 3446	. . . . . . . . . . . 12 ⊢ 𝑎 ∈ V
15	14	ideq 5809	. . . . . . . . . . 11 ⊢ (𝐴 I 𝑎 ↔ 𝐴 = 𝑎)
16		eqcom 2744	. . . . . . . . . . 11 ⊢ (𝐴 = 𝑎 ↔ 𝑎 = 𝐴)
17	15, 16	bitri 275	. . . . . . . . . 10 ⊢ (𝐴 I 𝑎 ↔ 𝑎 = 𝐴)
18		vex 3446	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
19	4, 18	brsingle 36131	. . . . . . . . . 10 ⊢ (𝐴Singleton𝑏 ↔ 𝑏 = {𝐴})
20	17, 19	anbi12i 629	. . . . . . . . 9 ⊢ ((𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ↔ (𝑎 = 𝐴 ∧ 𝑏 = {𝐴}))
21	20	anbi1i 625	. . . . . . . 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 1089	. . . . . . . 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 1851	. . . . 5 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ ∃𝑎∃𝑏(𝑎 = 𝐴 ∧ 𝑏 = {𝐴} ∧ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵)))
27		19.41vv 1952	. . . . 5 ⊢ (∃𝑎∃𝑏((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵) ↔ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎 ∧ 𝐴Singleton𝑏) ∧ 𝑥Cup𝐵))
28		opeq1 4831	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
29	28	eqeq2d 2748	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, 𝑏⟩))
30	29	anbi1d 632	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵)))
31		opeq2 4832	. . . . . . . 8 ⊢ (𝑏 = {𝐴} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐴}⟩)
32	31	eqeq2d 2748	. . . . . . 7 ⊢ (𝑏 = {𝐴} → (𝑥 = ⟨𝐴, 𝑏⟩ ↔ 𝑥 = ⟨𝐴, {𝐴}⟩))
33	32	anbi1d 632	. . . . . 6 ⊢ (𝑏 = {𝐴} → ((𝑥 = ⟨𝐴, 𝑏⟩ ∧ 𝑥Cup𝐵) ↔ (𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵)))
34	4, 5, 30, 33	ceqsex2v 3496	. . . . 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 1850	. 2 ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ ∃𝑥(𝑥 = ⟨𝐴, {𝐴}⟩ ∧ 𝑥Cup𝐵))
38		df-suc 6331	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
39	38	eqeq2i 2750	. 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901  {csn 4582  ⟨cop 4588   class class class wbr 5100   I cid 5526  suc csuc 6327   ⊗ ctxp 36044  Singletoncsingle 36052  Cupccup 36060

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-symdif 4207  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-1st 7943  df-2nd 7944  df-txp 36068  df-singleton 36076  df-cup 36083

This theorem is referenced by:  brsuccf  36156  dfsuccf2  36157