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Mirrors > Home > ILE Home > Th. List > nfop | GIF version |
Description: Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.) |
Ref | Expression |
---|---|
nfop.1 | ⊢ Ⅎ𝑥𝐴 |
nfop.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfop | ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-op 3536 | . 2 ⊢ 〈𝐴, 𝐵〉 = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
2 | nfop.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfel1 2292 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V |
4 | nfop.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfel1 2292 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V |
6 | 2 | nfsn 3583 | . . . . . 6 ⊢ Ⅎ𝑥{𝐴} |
7 | 2, 4 | nfpr 3573 | . . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
8 | 6, 7 | nfpr 3573 | . . . . 5 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}} |
9 | 8 | nfcri 2275 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}} |
10 | 3, 5, 9 | nf3an 1545 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}) |
11 | 10 | nfab 2286 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})} |
12 | 1, 11 | nfcxfr 2278 | 1 ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 962 ∈ wcel 1480 {cab 2125 Ⅎwnfc 2268 Vcvv 2686 {csn 3527 {cpr 3528 〈cop 3530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 |
This theorem is referenced by: nfopd 3722 moop2 4173 fliftfuns 5699 dfmpo 6120 qliftfuns 6513 xpf1o 6738 caucvgprprlemaddq 7516 nfseq 10228 txcnp 12440 cnmpt1t 12454 cnmpt2t 12462 |
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