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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 wwf. (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
opabn1stprc | ⊢ (∃𝑦𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3176 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
2 | 1 | biantrur 526 | . . . . . . 7 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
3 | 2 | opabbii 4649 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} |
4 | 3 | dmeqi 5247 | . . . . 5 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} |
5 | id 22 | . . . . . . 7 ⊢ (∃𝑦𝜑 → ∃𝑦𝜑) | |
6 | 5 | ralrimivw 2950 | . . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑) |
7 | dmopab3 5259 | . . . . . 6 ⊢ (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} = V) | |
8 | 6, 7 | sylib 207 | . . . . 5 ⊢ (∃𝑦𝜑 → dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} = V) |
9 | 4, 8 | syl5eq 2656 | . . . 4 ⊢ (∃𝑦𝜑 → dom {〈𝑥, 𝑦〉 ∣ 𝜑} = V) |
10 | vprc 4724 | . . . . 5 ⊢ ¬ V ∈ V | |
11 | 10 | a1i 11 | . . . 4 ⊢ (∃𝑦𝜑 → ¬ V ∈ V) |
12 | 9, 11 | eqneltrd 2707 | . . 3 ⊢ (∃𝑦𝜑 → ¬ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) |
13 | dmexg 6989 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V → dom {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) | |
14 | 12, 13 | nsyl 134 | . 2 ⊢ (∃𝑦𝜑 → ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) |
15 | df-nel 2783 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V ↔ ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) | |
16 | 14, 15 | sylibr 223 | 1 ⊢ (∃𝑦𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∉ wnel 2781 ∀wral 2896 Vcvv 3173 {copab 4642 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: griedg0prc 40488 |
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