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Theorem preqsnd 4783
Description: Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.) (Revised by AV, 13-Jun-2022.)
Hypotheses
Ref Expression
preqsnd.1 (𝜑𝐴 ∈ V)
preqsnd.2 (𝜑𝐵 ∈ V)
Assertion
Ref Expression
preqsnd (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))

Proof of Theorem preqsnd
StepHypRef Expression
1 preqsnd.1 . . . 4 (𝜑𝐴 ∈ V)
21adantl 482 . . 3 ((𝐶 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
3 preqsnd.2 . . . 4 (𝜑𝐵 ∈ V)
43adantl 482 . . 3 ((𝐶 ∈ V ∧ 𝜑) → 𝐵 ∈ V)
5 simpl 483 . . 3 ((𝐶 ∈ V ∧ 𝜑) → 𝐶 ∈ V)
6 dfsn2 4572 . . . . 5 {𝐶} = {𝐶, 𝐶}
76eqeq2i 2834 . . . 4 ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
8 preq12bg 4778 . . . . 5 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶))))
9 oridm 898 . . . . 5 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
108, 9syl6bb 288 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
117, 10syl5bb 284 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
122, 4, 5, 5, 11syl22anc 834 . 2 ((𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
13 snprc 4647 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1413biimpi 217 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
1514adantr 481 . . . . 5 ((¬ 𝐶 ∈ V ∧ 𝜑) → {𝐶} = ∅)
1615eqeq2d 2832 . . . 4 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = ∅))
17 prnzg 4707 . . . . . . 7 (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅)
18 eqneqall 3027 . . . . . . 7 ({𝐴, 𝐵} = ∅ → ({𝐴, 𝐵} ≠ ∅ → (𝐴 = 𝐶𝐵 = 𝐶)))
1917, 18syl5com 31 . . . . . 6 (𝐴 ∈ V → ({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 241 . . 3 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} → (𝐴 = 𝐶𝐵 = 𝐶)))
23 eleq1 2900 . . . . . . . . . 10 (𝐶 = 𝐴 → (𝐶 ∈ V ↔ 𝐴 ∈ V))
2423eqcoms 2829 . . . . . . . . 9 (𝐴 = 𝐶 → (𝐶 ∈ V ↔ 𝐴 ∈ V))
2524notbid 319 . . . . . . . 8 (𝐴 = 𝐶 → (¬ 𝐶 ∈ V ↔ ¬ 𝐴 ∈ V))
26 pm2.21 123 . . . . . . . 8 𝐴 ∈ V → (𝐴 ∈ V → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))
2725, 26syl6bi 254 . . . . . . 7 (𝐴 = 𝐶 → (¬ 𝐶 ∈ V → (𝐴 ∈ V → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶}))))
2827com13 88 . . . . . 6 (𝐴 ∈ V → (¬ 𝐶 ∈ V → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶}))))
291, 28syl 17 . . . . 5 (𝜑 → (¬ 𝐶 ∈ V → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶}))))
3029impcom 408 . . . 4 ((¬ 𝐶 ∈ V ∧ 𝜑) → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))
3130impd 411 . . 3 ((¬ 𝐶 ∈ V ∧ 𝜑) → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶}))
3222, 31impbid 213 . 2 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
3312, 32pm2.61ian 808 1 (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3016  Vcvv 3495  c0 4290  {csn 4559  {cpr 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3497  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4560  df-pr 4562
This theorem is referenced by:  prnesn  4784  preqsn  4786  opeqsng  5385  1loopgrnb0  27212  disjdifprg  30254
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