Mathbox for Steve Rodriguez |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > iso0 | Structured version Visualization version GIF version |
Description: The empty set is an 𝑅, 𝑆 isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.) |
Ref | Expression |
---|---|
iso0 | ⊢ ∅ Isom 𝑅, 𝑆 (∅, ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1o0 6736 | . 2 ⊢ ∅:∅–1-1-onto→∅ | |
2 | ral0 4440 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦)) | |
3 | df-isom 6427 | . 2 ⊢ (∅ Isom 𝑅, 𝑆 (∅, ∅) ↔ (∅:∅–1-1-onto→∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦)))) | |
4 | 1, 2, 3 | mpbir2an 707 | 1 ⊢ ∅ Isom 𝑅, 𝑆 (∅, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wral 3063 ∅c0 4253 class class class wbr 5070 –1-1-onto→wf1o 6417 ‘cfv 6418 Isom wiso 6419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-isom 6427 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |