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Mirrors > Home > MPE Home > Th. List > opabid | Structured version Visualization version GIF version |
Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
opabid | ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 4932 | . 2 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
2 | copsexg 4956 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | |
3 | 2 | bicomd 213 | . 2 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑)) |
4 | df-opab 4713 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
5 | 1, 3, 4 | elab2 3354 | 1 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 〈cop 4183 {copab 4712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 |
This theorem is referenced by: opelopabsb 4985 ssopab2b 5002 dmopab 5335 rnopab 5370 funopab 5923 opabiota 6261 fvopab5 6309 f1ompt 6382 ovid 6777 zfrep6 7134 enssdom 7980 omxpenlem 8061 infxpenlem 8836 canthwelem 9472 pospo 16973 2ndcdisj 21259 lgsquadlem1 25105 lgsquadlem2 25106 h2hlm 27837 opabdm 29423 opabrn 29424 fpwrelmap 29508 eulerpartlemgvv 30438 phpreu 33393 poimirlem26 33435 vvdifopab 34024 bropabid 34128 diclspsn 36483 areaquad 37802 sprsymrelf 41745 |
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