Theorem iso0 5940

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 5609	. 2 ⊢ ∅:∅–1-1-onto→∅
2		ral0 3593	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦))
3		df-isom 5326	. 2 ⊢ (∅ Isom 𝑅, 𝑆 (∅, ∅) ↔ (∅:∅–1-1-onto→∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦))))
4	1, 2, 3	mpbir2an 948	1 ⊢ ∅ Isom 𝑅, 𝑆 (∅, ∅)

Syntax hints:   ↔ wb 105  ∀wral 2508  ∅c0 3491   class class class wbr 4082  –1-1-onto→wf1o 5316  ‘cfv 5317   Isom wiso 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-isom 5326

This theorem is referenced by:  zfz1iso  11058