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Theorem bm2.5ii 4188
 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 ∣ y A yx})
Distinct variable group:   x,y,A

Proof of Theorem bm2.5ii
StepHypRef Expression
1 bm2.5ii.1 . . 3 A V
21ssonunii 4181 . 2 (A ⊆ On → A On)
3 unissb 3601 . . . . . 6 ( Axy A yx)
43a1i 9 . . . . 5 (x On → ( Axy A yx))
54rabbiia 2541 . . . 4 {x On ∣ Ax} = {x On ∣ y A yx}
65inteqi 3610 . . 3 {x On ∣ Ax} = {x On ∣ y A yx}
7 intmin 3626 . . 3 ( A On → {x On ∣ Ax} = A)
86, 7syl5reqr 2084 . 2 ( A On → A = {x On ∣ y A yx})
92, 8syl 14 1 (A ⊆ On → A = {x On ∣ y A yx})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {crab 2304  Vcvv 2551   ⊆ wss 2911  ∪ cuni 3571  ∩ cint 3606  Oncon0 4066 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-int 3607  df-tr 3846  df-iord 4069  df-on 4071 This theorem is referenced by: (None)
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