Theorem snopeqop 5525

Description: Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
snopeqop.a	⊢ 𝐴 ∈ V
snopeqop.b	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
snopeqop	⊢ ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}))

Proof of Theorem snopeqop

Step	Hyp	Ref	Expression
1		eqcom 2747	. . . . 5 ⊢ ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩})
2		opeqsng 5522	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩} ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶})))
3	2	ancoms 458	. . . . 5 ⊢ ((𝐷 ∈ V ∧ 𝐶 ∈ V) → (⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩} ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶})))
4	1, 3	bitrid 283	. . . 4 ⊢ ((𝐷 ∈ V ∧ 𝐶 ∈ V) → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶})))
5		snopeqop.a	. . . . . . 7 ⊢ 𝐴 ∈ V
6		snopeqop.b	. . . . . . 7 ⊢ 𝐵 ∈ V
7	5, 6	opeqsn 5523	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
8	7	a1i 11	. . . . 5 ⊢ ((𝐷 ∈ V ∧ 𝐶 ∈ V) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
9	8	anbi2d 629	. . . 4 ⊢ ((𝐷 ∈ V ∧ 𝐶 ∈ V) → ((𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))))
10		3anan12 1096	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
11	10	bicomi 224	. . . . 5 ⊢ ((𝐶 = 𝐷 ∧ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}))
12	11	a1i 11	. . . 4 ⊢ ((𝐷 ∈ V ∧ 𝐶 ∈ V) → ((𝐶 = 𝐷 ∧ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})))
13	4, 9, 12	3bitrd 305	. . 3 ⊢ ((𝐷 ∈ V ∧ 𝐶 ∈ V) → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})))
14		opprc2 4922	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → ⟨𝐶, 𝐷⟩ = ∅)
15	14	eqeq2d 2751	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ {⟨𝐴, 𝐵⟩} = ∅))
16		opex 5484	. . . . . . . 8 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
17	16	snnz 4801	. . . . . . 7 ⊢ {⟨𝐴, 𝐵⟩} ≠ ∅
18		eqneqall 2957	. . . . . . 7 ⊢ ({⟨𝐴, 𝐵⟩} = ∅ → ({⟨𝐴, 𝐵⟩} ≠ ∅ → (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})))
19	17, 18	mpi 20	. . . . . 6 ⊢ ({⟨𝐴, 𝐵⟩} = ∅ → (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}))
20	15, 19	biimtrdi 253	. . . . 5 ⊢ (¬ 𝐷 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})))
21	20	adantr 480	. . . 4 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})))
22		eleq1 2832	. . . . . . . . . 10 ⊢ (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
23	22	notbid 318	. . . . . . . . 9 ⊢ (𝐷 = 𝐶 → (¬ 𝐷 ∈ V ↔ ¬ 𝐶 ∈ V))
24	23	eqcoms 2748	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (¬ 𝐷 ∈ V ↔ ¬ 𝐶 ∈ V))
25		pm2.21 123	. . . . . . . 8 ⊢ (¬ 𝐶 ∈ V → (𝐶 ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
26	24, 25	biimtrdi 253	. . . . . . 7 ⊢ (𝐶 = 𝐷 → (¬ 𝐷 ∈ V → (𝐶 ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩)))
27	26	impd 410	. . . . . 6 ⊢ (𝐶 = 𝐷 → ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
28	27	3ad2ant2 1134	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}) → ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
29	28	com12 32	. . . 4 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}) → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
30	21, 29	impbid 212	. . 3 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})))
31	13, 30	pm2.61ian 811	. 2 ⊢ (𝐶 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})))
32		opprc1 4921	. . . . 5 ⊢ (¬ 𝐶 ∈ V → ⟨𝐶, 𝐷⟩ = ∅)
33	32	eqeq2d 2751	. . . 4 ⊢ (¬ 𝐶 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ {⟨𝐴, 𝐵⟩} = ∅))
34	33, 19	biimtrdi 253	. . 3 ⊢ (¬ 𝐶 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})))
35		eleq1 2832	. . . . . . 7 ⊢ (𝐶 = {𝐴} → (𝐶 ∈ V ↔ {𝐴} ∈ V))
36	35	notbid 318	. . . . . 6 ⊢ (𝐶 = {𝐴} → (¬ 𝐶 ∈ V ↔ ¬ {𝐴} ∈ V))
37		snex 5451	. . . . . . 7 ⊢ {𝐴} ∈ V
38	37	pm2.24i 150	. . . . . 6 ⊢ (¬ {𝐴} ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩)
39	36, 38	biimtrdi 253	. . . . 5 ⊢ (𝐶 = {𝐴} → (¬ 𝐶 ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
40	39	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}) → (¬ 𝐶 ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
41	40	com12 32	. . 3 ⊢ (¬ 𝐶 ∈ V → ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}) → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
42	34, 41	impbid 212	. 2 ⊢ (¬ 𝐶 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})))
43	31, 42	pm2.61i 182	1 ⊢ ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488  ∅c0 4352  {csn 4648  ⟨cop 4654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655

This theorem is referenced by:  snopeqopsnid  5528  funopsn  7182