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Theorem bm2.5ii 7171
 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 7152 . 2 (𝐴 ⊆ On → 𝐴 ∈ On)
3 unissb 4621 . . . . 5 ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
43rabbii 3325 . . . 4 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
54inteqi 4631 . . 3 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
6 intmin 4649 . . 3 ( 𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = 𝐴)
75, 6syl5reqr 2809 . 2 ( 𝐴 ∈ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
82, 7syl 17 1 (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054  Vcvv 3340   ⊆ wss 3715  ∪ cuni 4588  ∩ cint 4627  Oncon0 5884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888 This theorem is referenced by: (None)
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