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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 3497 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
2 | 1 | biantrur 533 | . . . . . . 7 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
3 | 2 | opabbii 5133 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} |
4 | 3 | dmeqi 5773 | . . . . 5 ⊢ dom {〈𝑥, 𝑦〉 ∣ 𝜑} = dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} |
5 | id 22 | . . . . . . 7 ⊢ (∃𝑦𝜑 → ∃𝑦𝜑) | |
6 | 5 | ralrimivw 3183 | . . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑) |
7 | dmopab3 5788 | . . . . . 6 ⊢ (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} = V) | |
8 | 6, 7 | sylib 220 | . . . . 5 ⊢ (∃𝑦𝜑 → dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} = V) |
9 | 4, 8 | syl5eq 2868 | . . . 4 ⊢ (∃𝑦𝜑 → dom {〈𝑥, 𝑦〉 ∣ 𝜑} = V) |
10 | vprc 5219 | . . . . 5 ⊢ ¬ V ∈ V | |
11 | 10 | a1i 11 | . . . 4 ⊢ (∃𝑦𝜑 → ¬ V ∈ V) |
12 | 9, 11 | eqneltrd 2932 | . . 3 ⊢ (∃𝑦𝜑 → ¬ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) |
13 | dmexg 7613 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V → dom {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) | |
14 | 12, 13 | nsyl 142 | . 2 ⊢ (∃𝑦𝜑 → ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) |
15 | df-nel 3124 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V ↔ ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) | |
16 | 14, 15 | sylibr 236 | 1 ⊢ (∃𝑦𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∉ wnel 3123 ∀wral 3138 Vcvv 3494 {copab 5128 dom cdm 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-nel 3124 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: griedg0prc 27046 |
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