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Theorem bm2.5ii 4268
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 4261 . 2 (𝐴 ⊆ On → 𝐴 ∈ On)
3 unissb 3651 . . . . . 6 ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
43a1i 9 . . . . 5 (𝑥 ∈ On → ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥))
54rabbiia 2596 . . . 4 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
65inteqi 3660 . . 3 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
7 intmin 3676 . . 3 ( 𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = 𝐴)
86, 7syl5reqr 2130 . 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 103   = wceq 1285  wcel 1434  wral 2353  {crab 2357  Vcvv 2610  wss 2982   cuni 3621   cint 3656  Oncon0 4146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-in 2988  df-ss 2995  df-uni 3622  df-int 3657  df-tr 3896  df-iord 4149  df-on 4151
This theorem is referenced by: (None)
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