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Theorem intid 3987
Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
Hypothesis
Ref Expression
intid.1 𝐴 ∈ V
Assertion
Ref Expression
intid {𝑥𝐴𝑥} = {𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem intid
StepHypRef Expression
1 intid.1 . . . 4 𝐴 ∈ V
2 snexgOLD 3962 . . . 4 (𝐴 ∈ V → {𝐴} ∈ V)
31, 2ax-mp 7 . . 3 {𝐴} ∈ V
4 eleq2 2117 . . . 4 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
51snid 3429 . . . 4 𝐴 ∈ {𝐴}
64, 5intmin3 3669 . . 3 ({𝐴} ∈ V → {𝑥𝐴𝑥} ⊆ {𝐴})
73, 6ax-mp 7 . 2 {𝑥𝐴𝑥} ⊆ {𝐴}
81elintab 3653 . . . 4 (𝐴 {𝑥𝐴𝑥} ↔ ∀𝑥(𝐴𝑥𝐴𝑥))
9 id 19 . . . 4 (𝐴𝑥𝐴𝑥)
108, 9mpgbir 1358 . . 3 𝐴 {𝑥𝐴𝑥}
11 snssi 3535 . . 3 (𝐴 {𝑥𝐴𝑥} → {𝐴} ⊆ {𝑥𝐴𝑥})
1210, 11ax-mp 7 . 2 {𝐴} ⊆ {𝑥𝐴𝑥}
137, 12eqssi 2988 1 {𝑥𝐴𝑥} = {𝐴}
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  {cab 2042  Vcvv 2574  wss 2944  {csn 3402   cint 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-int 3643
This theorem is referenced by: (None)
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