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Theorem bm2.5ii 4326
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 4319 . 2 (𝐴 ⊆ On → 𝐴 ∈ On)
3 unissb 3689 . . . . . 6 ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
43a1i 9 . . . . 5 (𝑥 ∈ On → ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥))
54rabbiia 2605 . . . 4 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
65inteqi 3698 . . 3 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
7 intmin 3714 . . 3 ( 𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = 𝐴)
86, 7syl5reqr 2136 . 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 104   = wceq 1290  wcel 1439  wral 2360  {crab 2364  Vcvv 2620  wss 3000   cuni 3659   cint 3694  Oncon0 4199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-in 3006  df-ss 3013  df-uni 3660  df-int 3695  df-tr 3943  df-iord 4202  df-on 4204
This theorem is referenced by: (None)
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