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Mirrors > Home > MPE Home > Th. List > Mathboxes > n0el2 | Structured version Visualization version GIF version |
Description: Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 31-Jan-2018.) |
Ref | Expression |
---|---|
n0el2 | ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmopab3 5788 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = 𝐴) | |
2 | n0el 4321 | . 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥) | |
3 | cnvepres 35570 | . . . 4 ⊢ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | |
4 | 3 | dmeqi 5773 | . . 3 ⊢ dom (◡ E ↾ 𝐴) = dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} |
5 | 4 | eqeq1i 2826 | . 2 ⊢ (dom (◡ E ↾ 𝐴) = 𝐴 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = 𝐴) |
6 | 1, 2, 5 | 3bitr4i 305 | 1 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3138 ∅c0 4291 {copab 5128 E cep 5464 ◡ccnv 5554 dom cdm 5555 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-res 5567 |
This theorem is referenced by: n0el3 35900 |
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