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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 6645 | . 2 ⊢ ∅:∅–1-1-onto→∅ | |
2 | ral0 4455 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦)) | |
3 | df-isom 6358 | . 2 ⊢ (∅ Isom 𝑅, 𝑆 (∅, ∅) ↔ (∅:∅–1-1-onto→∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦)))) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ ∅ Isom 𝑅, 𝑆 (∅, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wral 3138 ∅c0 4290 class class class wbr 5058 –1-1-onto→wf1o 6348 ‘cfv 6349 Isom wiso 6350 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-isom 6358 |
This theorem is referenced by: (None) |
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