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Theorem sbthcl 8198
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 8077 . 2 Rel ≈
2 inss1 3941 . . 3 ( ≼ ∩ ≼ ) ⊆ ≼
3 reldom 8078 . . 3 Rel ≼
4 relss 5315 . . 3 (( ≼ ∩ ≼ ) ⊆ ≼ → (Rel ≼ → Rel ( ≼ ∩ ≼ )))
52, 3, 4mp2 9 . 2 Rel ( ≼ ∩ ≼ )
6 brin 4812 . . 3 (𝑥( ≼ ∩ ≼ )𝑦 ↔ (𝑥𝑦𝑥𝑦))
7 vex 3307 . . . . 5 𝑥 ∈ V
8 vex 3307 . . . . 5 𝑦 ∈ V
97, 8brcnv 5412 . . . 4 (𝑥𝑦𝑦𝑥)
109anbi2i 732 . . 3 ((𝑥𝑦𝑥𝑦) ↔ (𝑥𝑦𝑦𝑥))
11 sbthb 8197 . . 3 ((𝑥𝑦𝑦𝑥) ↔ 𝑥𝑦)
126, 10, 113bitrri 287 . 2 (𝑥𝑦𝑥( ≼ ∩ ≼ )𝑦)
131, 5, 12eqbrriv 5324 1 ≈ = ( ≼ ∩ ≼ )
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1596  cin 3679  wss 3680   class class class wbr 4760  ccnv 5217  Rel wrel 5223  cen 8069  cdom 8070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-er 7862  df-en 8073  df-dom 8074
This theorem is referenced by:  dfsdom2  8199
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