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Theorem sbthcl 8289
Description: Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)
Assertion
Ref Expression
sbthcl ≈ = ( ≼ ∩ ≼ )

Proof of Theorem sbthcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relen 8165 . 2 Rel ≈
2 inss1 3992 . . 3 ( ≼ ∩ ≼ ) ⊆ ≼
3 reldom 8166 . . 3 Rel ≼
4 relss 5376 . . 3 (( ≼ ∩ ≼ ) ⊆ ≼ → (Rel ≼ → Rel ( ≼ ∩ ≼ )))
52, 3, 4mp2 9 . 2 Rel ( ≼ ∩ ≼ )
6 brin 4861 . . 3 (𝑥( ≼ ∩ ≼ )𝑦 ↔ (𝑥𝑦𝑥𝑦))
7 vex 3353 . . . . 5 𝑥 ∈ V
8 vex 3353 . . . . 5 𝑦 ∈ V
97, 8brcnv 5473 . . . 4 (𝑥𝑦𝑦𝑥)
109anbi2i 616 . . 3 ((𝑥𝑦𝑥𝑦) ↔ (𝑥𝑦𝑦𝑥))
11 sbthb 8288 . . 3 ((𝑥𝑦𝑦𝑥) ↔ 𝑥𝑦)
126, 10, 113bitrri 289 . 2 (𝑥𝑦𝑥( ≼ ∩ ≼ )𝑦)
131, 5, 12eqbrriv 5384 1 ≈ = ( ≼ ∩ ≼ )
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  cin 3731  wss 3732   class class class wbr 4809  ccnv 5276  Rel wrel 5282  cen 8157  cdom 8158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-er 7947  df-en 8161  df-dom 8162
This theorem is referenced by:  dfsdom2  8290
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