Theorem iso0 44427

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.

Step	Hyp	Ref	Expression
1		f1o0 6807	. 2 ⊢ ∅:∅–1-1-onto→∅
2		ral0 4448	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦))
3		df-isom 6497	. 2 ⊢ (∅ Isom 𝑅, 𝑆 (∅, ∅) ↔ (∅:∅–1-1-onto→∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦))))
4	1, 2, 3	mpbir2an 711	1 ⊢ ∅ Isom 𝑅, 𝑆 (∅, ∅)

Syntax hints:   ↔ wb 206  ∀wral 3048  ∅c0 4282   class class class wbr 5095  –1-1-onto→wf1o 6487  ‘cfv 6488   Isom wiso 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-isom 6497

This theorem is referenced by: (None)