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Theorem intpr 4662
Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
Hypotheses
Ref Expression
intpr.1 𝐴 ∈ V
intpr.2 𝐵 ∈ V
Assertion
Ref Expression
intpr {𝐴, 𝐵} = (𝐴𝐵)

Proof of Theorem intpr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1947 . . . 4 (∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
2 vex 3343 . . . . . . . 8 𝑦 ∈ V
32elpr 4343 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
43imbi1i 338 . . . . . 6 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦))
5 jaob 857 . . . . . 6 (((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
64, 5bitri 264 . . . . 5 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
76albii 1896 . . . 4 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
8 intpr.1 . . . . . 6 𝐴 ∈ V
98clel4 3482 . . . . 5 (𝑥𝐴 ↔ ∀𝑦(𝑦 = 𝐴𝑥𝑦))
10 intpr.2 . . . . . 6 𝐵 ∈ V
1110clel4 3482 . . . . 5 (𝑥𝐵 ↔ ∀𝑦(𝑦 = 𝐵𝑥𝑦))
129, 11anbi12i 735 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
131, 7, 123bitr4i 292 . . 3 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ (𝑥𝐴𝑥𝐵))
14 vex 3343 . . . 4 𝑥 ∈ V
1514elint 4633 . . 3 (𝑥 {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦))
16 elin 3939 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
1713, 15, 163bitr4i 292 . 2 (𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵))
1817eqriv 2757 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  wal 1630   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  {cpr 4323   cint 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-in 3722  df-sn 4322  df-pr 4324  df-int 4628
This theorem is referenced by:  intprg  4663  uniintsn  4666  op1stb  5088  fiint  8404  shincli  28551  chincli  28649
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