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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 6886 | . 2 ⊢ ∅:∅–1-1-onto→∅ | |
2 | ral0 4519 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦)) | |
3 | df-isom 6572 | . 2 ⊢ (∅ Isom 𝑅, 𝑆 (∅, ∅) ↔ (∅:∅–1-1-onto→∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦)))) | |
4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ ∅ Isom 𝑅, 𝑆 (∅, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∀wral 3059 ∅c0 4339 class class class wbr 5148 –1-1-onto→wf1o 6562 ‘cfv 6563 Isom wiso 6564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-isom 6572 |
This theorem is referenced by: (None) |
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