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Theorem preqsnd 4524
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 467 . . 3 ((𝐶 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
3 preqsnd.2 . . . 4 (𝜑𝐵 ∈ V)
43adantl 467 . . 3 ((𝐶 ∈ V ∧ 𝜑) → 𝐵 ∈ V)
5 simpl 468 . . 3 ((𝐶 ∈ V ∧ 𝜑) → 𝐶 ∈ V)
6 dfsn2 4330 . . . . 5 {𝐶} = {𝐶, 𝐶}
76eqeq2i 2783 . . . 4 ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
8 preq12bg 4518 . . . . 5 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶))))
9 oridm 882 . . . . 5 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
108, 9syl6bb 276 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
117, 10syl5bb 272 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
122, 4, 5, 5, 11syl22anc 1477 . 2 ((𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
13 snprc 4390 . . . . . . 7 𝐶 ∈ V ↔ {𝐶} = ∅)
1413biimpi 206 . . . . . 6 𝐶 ∈ V → {𝐶} = ∅)
1514adantr 466 . . . . 5 ((¬ 𝐶 ∈ V ∧ 𝜑) → {𝐶} = ∅)
1615eqeq2d 2781 . . . 4 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = ∅))
17 prnzg 4447 . . . . . . 7 (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅)
18 eqneqall 2954 . . . . . . 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 467 . . . 4 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = ∅ → (𝐴 = 𝐶𝐵 = 𝐶)))
2216, 21sylbid 230 . . 3 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} → (𝐴 = 𝐶𝐵 = 𝐶)))
23 eleq1 2838 . . . . . . . . . 10 (𝐶 = 𝐴 → (𝐶 ∈ V ↔ 𝐴 ∈ V))
2423eqcoms 2779 . . . . . . . . 9 (𝐴 = 𝐶 → (𝐶 ∈ V ↔ 𝐴 ∈ V))
2524notbid 307 . . . . . . . 8 (𝐴 = 𝐶 → (¬ 𝐶 ∈ V ↔ ¬ 𝐴 ∈ V))
26 pm2.21 121 . . . . . . . 8 𝐴 ∈ V → (𝐴 ∈ V → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))
2725, 26syl6bi 243 . . . . . . 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 394 . . . 4 ((¬ 𝐶 ∈ V ∧ 𝜑) → (𝐴 = 𝐶 → (𝐵 = 𝐶 → {𝐴, 𝐵} = {𝐶})))
3130impd 396 . . 3 ((¬ 𝐶 ∈ V ∧ 𝜑) → ((𝐴 = 𝐶𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶}))
3222, 31impbid 202 . 2 ((¬ 𝐶 ∈ V ∧ 𝜑) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
3312, 32pm2.61ian 806 1 (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 828   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  c0 4064  {csn 4317  {cpr 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3727  df-un 3729  df-nul 4065  df-sn 4318  df-pr 4320
This theorem is referenced by:  prnesn  4526  preqsn  4528  opeqsng  5095  1loopgrnb0  26634  disjdifprg  29727
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