![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > griedg0prc | Structured version Visualization version GIF version |
Description: The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
griedg0prc.u | ⊢ 𝑈 = {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} |
Ref | Expression |
---|---|
griedg0prc | ⊢ 𝑈 ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5062 | . . . 4 ⊢ ∅ ∈ V | |
2 | feq1 6319 | . . . 4 ⊢ (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅)) | |
3 | f0 6383 | . . . 4 ⊢ ∅:∅⟶∅ | |
4 | 1, 2, 3 | ceqsexv2d 3457 | . . 3 ⊢ ∃𝑒 𝑒:∅⟶∅ |
5 | opabn1stprc 7557 | . . 3 ⊢ (∃𝑒 𝑒:∅⟶∅ → {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} ∉ V) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} ∉ V |
7 | griedg0prc.u | . . 3 ⊢ 𝑈 = {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} | |
8 | neleq1 3072 | . . 3 ⊢ (𝑈 = {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} → (𝑈 ∉ V ↔ {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} ∉ V)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ (𝑈 ∉ V ↔ {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} ∉ V) |
10 | 6, 9 | mpbir 223 | 1 ⊢ 𝑈 ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 ∃wex 1742 ∉ wnel 3067 Vcvv 3409 ∅c0 4173 {copab 4985 ⟶wf 6178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-nel 3068 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-fun 6184 df-fn 6185 df-f 6186 |
This theorem is referenced by: usgrprc 26741 rgrusgrprc 27064 |
Copyright terms: Public domain | W3C validator |