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Mirrors > Home > MPE Home > Th. List > opabn1stprc | Structured version Visualization version GIF version |
Description: An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wff. (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
opabn1stprc | ⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3473 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
2 | 1 | biantrur 530 | . . . . . . 7 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
3 | 2 | opabbii 5209 | . . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} |
4 | 3 | dmeqi 5901 | . . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} |
5 | id 22 | . . . . . . 7 ⊢ (∃𝑦𝜑 → ∃𝑦𝜑) | |
6 | 5 | ralrimivw 3145 | . . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑) |
7 | dmopab3 5916 | . . . . . 6 ⊢ (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V) | |
8 | 6, 7 | sylib 217 | . . . . 5 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V) |
9 | 4, 8 | eqtrid 2779 | . . . 4 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = V) |
10 | vprc 5309 | . . . . 5 ⊢ ¬ V ∈ V | |
11 | 10 | a1i 11 | . . . 4 ⊢ (∃𝑦𝜑 → ¬ V ∈ V) |
12 | 9, 11 | eqneltrd 2848 | . . 3 ⊢ (∃𝑦𝜑 → ¬ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V) |
13 | dmexg 7901 | . . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V) | |
14 | 12, 13 | nsyl 140 | . 2 ⊢ (∃𝑦𝜑 → ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V) |
15 | df-nel 3042 | . 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V) | |
16 | 14, 15 | sylibr 233 | 1 ⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∉ wnel 3041 ∀wral 3056 Vcvv 3469 {copab 5204 dom cdm 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-nel 3042 df-ral 3057 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-cnv 5680 df-dm 5682 df-rn 5683 |
This theorem is referenced by: griedg0prc 29051 |
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