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Theorem bm2.5ii 4513
Description: Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
Hypothesis
Ref Expression
bm2.5ii.1 𝐴 ∈ V
Assertion
Ref Expression
bm2.5ii (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem bm2.5ii
StepHypRef Expression
1 bm2.5ii.1 . . 3 𝐴 ∈ V
21ssonunii 4506 . 2 (𝐴 ⊆ On → 𝐴 ∈ On)
3 intmin 3879 . . 3 ( 𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = 𝐴)
4 unissb 3854 . . . . . 6 ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
54a1i 9 . . . . 5 (𝑥 ∈ On → ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥))
65rabbiia 2737 . . . 4 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
76inteqi 3863 . . 3 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
83, 7eqtr3di 2237 . 2 ( 𝐴 ∈ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
92, 8syl 14 1 (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wcel 2160  wral 2468  {crab 2472  Vcvv 2752  wss 3144   cuni 3824   cint 3859  Oncon0 4381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-int 3860  df-tr 4117  df-iord 4384  df-on 4386
This theorem is referenced by: (None)
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