Theorem bj-elid3 33398

Description: Characterization of the elements of I. (Contributed by BJ, 29-Mar-2020.)

Assertion

Ref	Expression
bj-elid3	⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-elid3

Step	Hyp	Ref	Expression
1		bj-elid 33396	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
2		opelxp 5303	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	anbi1i 733	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
4		op1stg 7345	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
5		op2ndg 7346	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
6	4, 5	eqeq12d 2775	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩) ↔ 𝐴 = 𝐵))
7	6	pm5.32i 672	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
8		simpl 474	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
9	8	anim1i 593	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐴 = 𝐵))
10		simpl 474	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
11		eleq1 2827	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
12	11	biimpac 504	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
13		simpr 479	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
14	10, 12, 13	jca31 558	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
15	9, 14	impbii 199	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
16	7, 15	bitri 264	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
17	3, 16	bitri 264	. 2 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
18	1, 17	bitri 264	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327   I cid 5173   × cxp 5264  ‘cfv 6049  1^st c1st 7331  2^nd c2nd 7332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-1st 7333  df-2nd 7334

This theorem is referenced by:  bj-eldiag2  33403