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Theorem moop2 3979
Description: "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
moop2.1 B V
Assertion
Ref Expression
moop2 ∃*x A = ⟨B, x
Distinct variable group:   x,A
Allowed substitution hint:   B(x)

Proof of Theorem moop2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqtr2 2055 . . . 4 ((A = ⟨B, x A = ⟨y / xB, y⟩) → ⟨B, x⟩ = ⟨y / xB, y⟩)
2 moop2.1 . . . . . 6 B V
3 vex 2554 . . . . . 6 x V
42, 3opth 3965 . . . . 5 (⟨B, x⟩ = ⟨y / xB, y⟩ ↔ (B = y / xB x = y))
54simprbi 260 . . . 4 (⟨B, x⟩ = ⟨y / xB, y⟩ → x = y)
61, 5syl 14 . . 3 ((A = ⟨B, x A = ⟨y / xB, y⟩) → x = y)
76gen2 1336 . 2 xy((A = ⟨B, x A = ⟨y / xB, y⟩) → x = y)
8 nfcsb1v 2876 . . . . 5 xy / xB
9 nfcv 2175 . . . . 5 xy
108, 9nfop 3556 . . . 4 xy / xB, y
1110nfeq2 2186 . . 3 x A = ⟨y / xB, y
12 csbeq1a 2854 . . . . 5 (x = yB = y / xB)
13 id 19 . . . . 5 (x = yx = y)
1412, 13opeq12d 3548 . . . 4 (x = y → ⟨B, x⟩ = ⟨y / xB, y⟩)
1514eqeq2d 2048 . . 3 (x = y → (A = ⟨B, x⟩ ↔ A = ⟨y / xB, y⟩))
1611, 15mo4f 1957 . 2 (∃*x A = ⟨B, x⟩ ↔ xy((A = ⟨B, x A = ⟨y / xB, y⟩) → x = y))
177, 16mpbir 134 1 ∃*x A = ⟨B, x
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242   wcel 1390  ∃*wmo 1898  Vcvv 2551  csb 2846  cop 3370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by: (None)
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