Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  bm2.5ii GIF version

Theorem bm2.5ii 4222
 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 4215 . 2 (𝐴 ⊆ On → 𝐴 ∈ On)
3 unissb 3610 . . . . . 6 ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
43a1i 9 . . . . 5 (𝑥 ∈ On → ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥))
54rabbiia 2547 . . . 4 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
65inteqi 3619 . . 3 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
7 intmin 3635 . . 3 ( 𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = 𝐴)
86, 7syl5reqr 2087 . 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 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310  Vcvv 2557   ⊆ wss 2917  ∪ cuni 3580  ∩ cint 3615  Oncon0 4100 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator