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Theorem preqsnd 4795
Description: Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.) (Revised by AV, 13-Jun-2022.) (Revised by AV, 16-Aug-2024.)
Hypotheses
Ref Expression
preqsnd.1 (𝜑𝐴𝑉)
preqsnd.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
preqsnd (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))

Proof of Theorem preqsnd
StepHypRef Expression
1 preqsnd.1 . . . 4 (𝜑𝐴𝑉)
21adantl 482 . . 3 ((𝐶 ∈ V ∧ 𝜑) → 𝐴𝑉)
3 preqsnd.2 . . . 4 (𝜑𝐵𝑊)
43adantl 482 . . 3 ((𝐶 ∈ V ∧ 𝜑) → 𝐵𝑊)
5 simpl 483 . . 3 ((𝐶 ∈ V ∧ 𝜑) → 𝐶 ∈ V)
6 dfsn2 4580 . . . . 5 {𝐶} = {𝐶, 𝐶}
76eqeq2i 2753 . . . 4 ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
8 preq12bg 4790 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶))))
9 oridm 902 . . . . 5 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
108, 9bitrdi 287 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
117, 10bitrid 282 . . 3 (((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
122, 4, 5, 5, 11syl22anc 836 . 2 ((𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
13 snprc 4659 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1413biimpi 215 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
1514adantr 481 . . . . 5 ((¬ 𝐶 ∈ V ∧ 𝜑) → {𝐶} = ∅)
1615eqeq2d 2751 . . . 4 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = ∅))
17 prnzg 4720 . . . . . . 7 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
18 eqneqall 2956 . . . . . . 7 ({𝐴, 𝐵} = ∅ → ({𝐴, 𝐵} ≠ ∅ → (𝐴 = 𝐶𝐵 = 𝐶)))
1917, 18syl5com 31 . . . . . 6 (𝐴𝑉 → ({𝐴, 𝐵} = ∅ → (𝐴 = 𝐶𝐵 = 𝐶)))
201, 19syl 17 . . . . 5 (𝜑 → ({𝐴, 𝐵} = ∅ → (𝐴 = 𝐶𝐵 = 𝐶)))
2120adantl 482 . . . 4 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = ∅ → (𝐴 = 𝐶𝐵 = 𝐶)))
2216, 21sylbid 239 . . 3 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} → (𝐴 = 𝐶𝐵 = 𝐶)))
23 eleq1 2828 . . . . . . . . . 10 (𝐶 = 𝐴 → (𝐶 ∈ V ↔ 𝐴 ∈ V))
2423eqcoms 2748 . . . . . . . . 9 (𝐴 = 𝐶 → (𝐶 ∈ V ↔ 𝐴 ∈ V))
2524notbid 318 . . . . . . . 8 (𝐴 = 𝐶 → (¬ 𝐶 ∈ V ↔ ¬ 𝐴 ∈ V))
26 pm2.24 124 . . . . . . . . 9 (𝐴 ∈ V → (¬ 𝐴 ∈ V → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))
27 elex 3449 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ V)
2826, 27syl11 33 . . . . . . . 8 𝐴 ∈ V → (𝐴𝑉 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))
2925, 28syl6bi 252 . . . . . . 7 (𝐴 = 𝐶 → (¬ 𝐶 ∈ V → (𝐴𝑉 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶}))))
3029com13 88 . . . . . 6 (𝐴𝑉 → (¬ 𝐶 ∈ V → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶}))))
311, 30syl 17 . . . . 5 (𝜑 → (¬ 𝐶 ∈ V → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶}))))
3231impcom 408 . . . 4 ((¬ 𝐶 ∈ V ∧ 𝜑) → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))
3332impd 411 . . 3 ((¬ 𝐶 ∈ V ∧ 𝜑) → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶}))
3422, 33impbid 211 . 2 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
3512, 34pm2.61ian 809 1 (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1542  wcel 2110  wne 2945  Vcvv 3431  c0 4262  {csn 4567  {cpr 4569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263  df-sn 4568  df-pr 4570
This theorem is referenced by:  prnesn  4796  preqsn  4798  opeqsng  5421  1loopgrnb0  27880  disjdifprg  30923
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