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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 3440 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
2 | 1 | biantrur 531 | . . . . . . 7 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
3 | 2 | opabbii 5029 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} |
4 | 3 | dmeqi 5659 | . . . . 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 5674 | . . . . . 6 ⊢ (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} = V) | |
8 | 6, 7 | sylib 219 | . . . . 5 ⊢ (∃𝑦𝜑 → dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝜑)} = V) |
9 | 4, 8 | syl5eq 2843 | . . . 4 ⊢ (∃𝑦𝜑 → dom {〈𝑥, 𝑦〉 ∣ 𝜑} = V) |
10 | vprc 5110 | . . . . 5 ⊢ ¬ V ∈ V | |
11 | 10 | a1i 11 | . . . 4 ⊢ (∃𝑦𝜑 → ¬ V ∈ V) |
12 | 9, 11 | eqneltrd 2902 | . . 3 ⊢ (∃𝑦𝜑 → ¬ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) |
13 | dmexg 7469 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V → dom {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) | |
14 | 12, 13 | nsyl 142 | . 2 ⊢ (∃𝑦𝜑 → ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) |
15 | df-nel 3091 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V ↔ ¬ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ V) | |
16 | 14, 15 | sylibr 235 | 1 ⊢ (∃𝑦𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 ∉ wnel 3090 ∀wral 3105 Vcvv 3437 {copab 5024 dom cdm 5443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-nel 3091 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-cnv 5451 df-dm 5453 df-rn 5454 |
This theorem is referenced by: griedg0prc 26729 |
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