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Mirrors > Home > MPE Home > Th. List > Mathboxes > n0elqs2 | Structured version Visualization version GIF version |
Description: Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 25-Jul-2021.) |
Ref | Expression |
---|---|
n0elqs2 | ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ dom (𝑅 ↾ 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0elqs 34098 | . 2 ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ 𝐴 ⊆ dom 𝑅) | |
2 | ssdmres 5420 | . 2 ⊢ (𝐴 ⊆ dom 𝑅 ↔ dom (𝑅 ↾ 𝐴) = 𝐴) | |
3 | 1, 2 | bitri 264 | 1 ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ dom (𝑅 ↾ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∅c0 3915 dom cdm 5114 ↾ cres 5116 / cqs 7741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ec 7744 df-qs 7748 |
This theorem is referenced by: (None) |
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