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Theorem snopeqop 5362
 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 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))

Proof of Theorem snopeqop
StepHypRef Expression
1 eqcom 2805 . . . . 5 ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩})
2 opeqsng 5359 . . . . . 6 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩} ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶})))
32ancoms 462 . . . . 5 ((𝐷 ∈ V ∧ 𝐶 ∈ V) → (⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩} ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶})))
41, 3syl5bb 286 . . . 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
75, 6opeqsn 5360 . . . . . 6 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
87a1i 11 . . . . 5 ((𝐷 ∈ V ∧ 𝐶 ∈ V) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
98anbi2d 631 . . . 4 ((𝐷 ∈ V ∧ 𝐶 ∈ V) → ((𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵𝐶 = {𝐴}))))
10 3anan12 1093 . . . . . 6 ((𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵𝐶 = {𝐴})))
1110bicomi 227 . . . . 5 ((𝐶 = 𝐷 ∧ (𝐴 = 𝐵𝐶 = {𝐴})) ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))
1211a1i 11 . . . 4 ((𝐷 ∈ V ∧ 𝐶 ∈ V) → ((𝐶 = 𝐷 ∧ (𝐴 = 𝐵𝐶 = {𝐴})) ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴})))
134, 9, 123bitrd 308 . . 3 ((𝐷 ∈ V ∧ 𝐶 ∈ V) → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴})))
14 opprc2 4791 . . . . . . 7 𝐷 ∈ V → ⟨𝐶, 𝐷⟩ = ∅)
1514eqeq2d 2809 . . . . . 6 𝐷 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ {⟨𝐴, 𝐵⟩} = ∅))
16 opex 5322 . . . . . . . 8 𝐴, 𝐵⟩ ∈ V
1716snnz 4672 . . . . . . 7 {⟨𝐴, 𝐵⟩} ≠ ∅
18 eqneqall 2998 . . . . . . 7 ({⟨𝐴, 𝐵⟩} = ∅ → ({⟨𝐴, 𝐵⟩} ≠ ∅ → (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴})))
1917, 18mpi 20 . . . . . 6 ({⟨𝐴, 𝐵⟩} = ∅ → (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))
2015, 19syl6bi 256 . . . . 5 𝐷 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴})))
2120adantr 484 . . . 4 ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴})))
22 eleq1 2877 . . . . . . . . . 10 (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
2322notbid 321 . . . . . . . . 9 (𝐷 = 𝐶 → (¬ 𝐷 ∈ V ↔ ¬ 𝐶 ∈ V))
2423eqcoms 2806 . . . . . . . 8 (𝐶 = 𝐷 → (¬ 𝐷 ∈ V ↔ ¬ 𝐶 ∈ V))
25 pm2.21 123 . . . . . . . 8 𝐶 ∈ V → (𝐶 ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
2624, 25syl6bi 256 . . . . . . 7 (𝐶 = 𝐷 → (¬ 𝐷 ∈ V → (𝐶 ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩)))
2726impd 414 . . . . . 6 (𝐶 = 𝐷 → ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
28273ad2ant2 1131 . . . . 5 ((𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}) → ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
2928com12 32 . . . 4 ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → ((𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}) → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
3021, 29impbid 215 . . 3 ((¬ 𝐷 ∈ V ∧ 𝐶 ∈ V) → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴})))
3113, 30pm2.61ian 811 . 2 (𝐶 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴})))
32 opprc1 4790 . . . . 5 𝐶 ∈ V → ⟨𝐶, 𝐷⟩ = ∅)
3332eqeq2d 2809 . . . 4 𝐶 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ {⟨𝐴, 𝐵⟩} = ∅))
3433, 19syl6bi 256 . . 3 𝐶 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴})))
35 eleq1 2877 . . . . . . 7 (𝐶 = {𝐴} → (𝐶 ∈ V ↔ {𝐴} ∈ V))
3635notbid 321 . . . . . 6 (𝐶 = {𝐴} → (¬ 𝐶 ∈ V ↔ ¬ {𝐴} ∈ V))
37 snex 5298 . . . . . . 7 {𝐴} ∈ V
3837pm2.24i 153 . . . . . 6 (¬ {𝐴} ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩)
3936, 38syl6bi 256 . . . . 5 (𝐶 = {𝐴} → (¬ 𝐶 ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
40393ad2ant3 1132 . . . 4 ((𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}) → (¬ 𝐶 ∈ V → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
4140com12 32 . . 3 𝐶 ∈ V → ((𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}) → {⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩))
4234, 41impbid 215 . 2 𝐶 ∈ V → ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴})))
4331, 42pm2.61i 185 1 ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441  ∅c0 4243  {csn 4525  ⟨cop 4531 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532 This theorem is referenced by:  snopeqopsnid  5365  funopsn  6888
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