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Theorem sbthcl 8627
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 8501 . 2 Rel ≈
2 inss1 4158 . . 3 ( ≼ ∩ ≼ ) ⊆ ≼
3 reldom 8502 . . 3 Rel ≼
4 relss 5624 . . 3 (( ≼ ∩ ≼ ) ⊆ ≼ → (Rel ≼ → Rel ( ≼ ∩ ≼ )))
52, 3, 4mp2 9 . 2 Rel ( ≼ ∩ ≼ )
6 brin 5085 . . 3 (𝑥( ≼ ∩ ≼ )𝑦 ↔ (𝑥𝑦𝑥𝑦))
7 vex 3447 . . . . 5 𝑥 ∈ V
8 vex 3447 . . . . 5 𝑦 ∈ V
97, 8brcnv 5721 . . . 4 (𝑥𝑦𝑦𝑥)
109anbi2i 625 . . 3 ((𝑥𝑦𝑥𝑦) ↔ (𝑥𝑦𝑦𝑥))
11 sbthb 8626 . . 3 ((𝑥𝑦𝑦𝑥) ↔ 𝑥𝑦)
126, 10, 113bitrri 301 . 2 (𝑥𝑦𝑥( ≼ ∩ ≼ )𝑦)
131, 5, 12eqbrriv 5632 1 ≈ = ( ≼ ∩ ≼ )
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  cin 3883  wss 3884   class class class wbr 5033  ccnv 5522  Rel wrel 5528  cen 8493  cdom 8494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-er 8276  df-en 8497  df-dom 8498
This theorem is referenced by:  dfsdom2  8628
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