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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 5300 | . . . 4 ⊢ ∅ ∈ V | |
2 | feq1 6685 | . . . 4 ⊢ (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅)) | |
3 | f0 6759 | . . . 4 ⊢ ∅:∅⟶∅ | |
4 | 1, 2, 3 | ceqsexv2d 3525 | . . 3 ⊢ ∃𝑒 𝑒:∅⟶∅ |
5 | opabn1stprc 8026 | . . 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 3051 | . . 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 230 | 1 ⊢ 𝑈 ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∃wex 1781 ∉ wnel 3045 Vcvv 3473 ∅c0 4318 {copab 5203 ⟶wf 6528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-fun 6534 df-fn 6535 df-f 6536 |
This theorem is referenced by: usgrprc 28388 rgrusgrprc 28711 |
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