Theorem el1st 4729

Description: Membership in 1^st. (Contributed by SF, 5-Jan-2015.)

Assertion

Ref	Expression
el1st	⊢ (A ∈ 1st ↔ ∃x∃y A = ⟨⟨x, y⟩, x⟩)

Distinct variable group:   x,A,y

Proof of Theorem el1st

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-1st 4723	. . . 4 ⊢ 1st = {⟨z, x⟩ ∣ ∃y z = ⟨x, y⟩}
2	1	eleq2i 2417	. . 3 ⊢ (A ∈ 1st ↔ A ∈ {⟨z, x⟩ ∣ ∃y z = ⟨x, y⟩})
3		elopab 4696	. . 3 ⊢ (A ∈ {⟨z, x⟩ ∣ ∃y z = ⟨x, y⟩} ↔ ∃z∃x(A = ⟨z, x⟩ ∧ ∃y z = ⟨x, y⟩))
4	2, 3	bitri 240	. 2 ⊢ (A ∈ 1st ↔ ∃z∃x(A = ⟨z, x⟩ ∧ ∃y z = ⟨x, y⟩))
5		excom 1741	. . 3 ⊢ (∃z∃x(A = ⟨z, x⟩ ∧ ∃y z = ⟨x, y⟩) ↔ ∃x∃z(A = ⟨z, x⟩ ∧ ∃y z = ⟨x, y⟩))
6		excom 1741	. . . . 5 ⊢ (∃y∃z(A = ⟨z, x⟩ ∧ z = ⟨x, y⟩) ↔ ∃z∃y(A = ⟨z, x⟩ ∧ z = ⟨x, y⟩))
7		exancom 1586	. . . . . . 7 ⊢ (∃z(A = ⟨z, x⟩ ∧ z = ⟨x, y⟩) ↔ ∃z(z = ⟨x, y⟩ ∧ A = ⟨z, x⟩))
8		vex 2862	. . . . . . . . 9 ⊢ x ∈ V
9		vex 2862	. . . . . . . . 9 ⊢ y ∈ V
10	8, 9	opex 4588	. . . . . . . 8 ⊢ ⟨x, y⟩ ∈ V
11		opeq1 4578	. . . . . . . . 9 ⊢ (z = ⟨x, y⟩ → ⟨z, x⟩ = ⟨⟨x, y⟩, x⟩)
12	11	eqeq2d 2364	. . . . . . . 8 ⊢ (z = ⟨x, y⟩ → (A = ⟨z, x⟩ ↔ A = ⟨⟨x, y⟩, x⟩))
13	10, 12	ceqsexv 2894	. . . . . . 7 ⊢ (∃z(z = ⟨x, y⟩ ∧ A = ⟨z, x⟩) ↔ A = ⟨⟨x, y⟩, x⟩)
14	7, 13	bitri 240	. . . . . 6 ⊢ (∃z(A = ⟨z, x⟩ ∧ z = ⟨x, y⟩) ↔ A = ⟨⟨x, y⟩, x⟩)
15	14	exbii 1582	. . . . 5 ⊢ (∃y∃z(A = ⟨z, x⟩ ∧ z = ⟨x, y⟩) ↔ ∃y A = ⟨⟨x, y⟩, x⟩)
16		exdistr 1906	. . . . 5 ⊢ (∃z∃y(A = ⟨z, x⟩ ∧ z = ⟨x, y⟩) ↔ ∃z(A = ⟨z, x⟩ ∧ ∃y z = ⟨x, y⟩))
17	6, 15, 16	3bitr3ri 267	. . . 4 ⊢ (∃z(A = ⟨z, x⟩ ∧ ∃y z = ⟨x, y⟩) ↔ ∃y A = ⟨⟨x, y⟩, x⟩)
18	17	exbii 1582	. . 3 ⊢ (∃x∃z(A = ⟨z, x⟩ ∧ ∃y z = ⟨x, y⟩) ↔ ∃x∃y A = ⟨⟨x, y⟩, x⟩)
19	5, 18	bitri 240	. 2 ⊢ (∃z∃x(A = ⟨z, x⟩ ∧ ∃y z = ⟨x, y⟩) ↔ ∃x∃y A = ⟨⟨x, y⟩, x⟩)
20	4, 19	bitri 240	1 ⊢ (A ∈ 1st ↔ ∃x∃y A = ⟨⟨x, y⟩, x⟩)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561  {copab 4622  1^st c1st 4717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-1st 4723

This theorem is referenced by:  br1stg  4730