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Theorem iso0 43555
Description: The empty set is an 𝑅, 𝑆 isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
Assertion
Ref Expression
iso0 ∅ Isom 𝑅, 𝑆 (∅, ∅)

Proof of Theorem iso0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1o0 6860 . 2 ∅:∅–1-1-onto→∅
2 ral0 4504 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦))
3 df-isom 6542 . 2 (∅ Isom 𝑅, 𝑆 (∅, ∅) ↔ (∅:∅–1-1-onto→∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦))))
41, 2, 3mpbir2an 708 1 ∅ Isom 𝑅, 𝑆 (∅, ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wral 3053  c0 4314   class class class wbr 5138  1-1-ontowf1o 6532  cfv 6533   Isom wiso 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-isom 6542
This theorem is referenced by: (None)
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