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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 6644 | . 2 ⊢ ∅:∅–1-1-onto→∅ | |
2 | ral0 4452 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦)) | |
3 | df-isom 6357 | . 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 207 ∀wral 3135 ∅c0 4288 class class class wbr 5057 –1-1-onto→wf1o 6347 ‘cfv 6348 Isom wiso 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-isom 6357 |
This theorem is referenced by: (None) |
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