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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 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 2 | 1 | biantrur 530 | . . . . . . 7 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
| 3 | 2 | opabbii 5210 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} |
| 4 | 3 | dmeqi 5915 | . . . . 5 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} |
| 5 | id 22 | . . . . . . 7 ⊢ (∃𝑦𝜑 → ∃𝑦𝜑) | |
| 6 | 5 | ralrimivw 3150 | . . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑) |
| 7 | dmopab3 5930 | . . . . . 6 ⊢ (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} = V) | |
| 8 | 6, 7 | sylib 218 | . . . . 5 ⊢ (∃𝑦𝜑 → dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} = V) |
| 9 | 4, 8 | eqtrid 2789 | . . . 4 ⊢ (∃𝑦𝜑 → dom {〈𝑥, 𝑦〉 ∣ 𝜑} = V) |
| 10 | vprc 5315 | . . . . 5 ⊢ ¬ V ∈ V | |
| 11 | 10 | a1i 11 | . . . 4 ⊢ (∃𝑦𝜑 → ¬ V ∈ V) |
| 12 | 9, 11 | eqneltrd 2861 | . . 3 ⊢ (∃𝑦𝜑 → ¬ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) |
| 13 | dmexg 7923 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V → dom {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) | |
| 14 | 12, 13 | nsyl 140 | . 2 ⊢ (∃𝑦𝜑 → ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) |
| 15 | df-nel 3047 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V ↔ ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) | |
| 16 | 14, 15 | sylibr 234 | 1 ⊢ (∃𝑦𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∉ wnel 3046 ∀wral 3061 Vcvv 3480 {copab 5205 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-nel 3047 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: griedg0prc 29281 |
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