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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 5202 | . . . 4 ⊢ ∅ ∈ V | |
2 | feq1 6488 | . . . 4 ⊢ (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅)) | |
3 | f0 6553 | . . . 4 ⊢ ∅:∅⟶∅ | |
4 | 1, 2, 3 | ceqsexv2d 3540 | . . 3 ⊢ ∃𝑒 𝑒:∅⟶∅ |
5 | opabn1stprc 7745 | . . 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 3125 | . . 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 232 | 1 ⊢ 𝑈 ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∃wex 1771 ∉ wnel 3120 Vcvv 3492 ∅c0 4288 {copab 5119 ⟶wf 6344 |
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 ax-un 7450 |
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-nel 3121 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-uni 4831 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 |
This theorem is referenced by: usgrprc 26975 rgrusgrprc 27298 |
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