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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 3483 | . 2 ⊢ 〈𝐴, 𝐵〉 = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
2 | nfop.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfel1 2251 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V |
4 | nfop.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfel1 2251 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V |
6 | 2 | nfsn 3530 | . . . . . 6 ⊢ Ⅎ𝑥{𝐴} |
7 | 2, 4 | nfpr 3520 | . . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
8 | 6, 7 | nfpr 3520 | . . . . 5 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}} |
9 | 8 | nfcri 2234 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}} |
10 | 3, 5, 9 | nf3an 1513 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}) |
11 | 10 | nfab 2245 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})} |
12 | 1, 11 | nfcxfr 2237 | 1 ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 930 ∈ wcel 1448 {cab 2086 Ⅎwnfc 2227 Vcvv 2641 {csn 3474 {cpr 3475 〈cop 3477 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-un 3025 df-sn 3480 df-pr 3481 df-op 3483 |
This theorem is referenced by: nfopd 3669 moop2 4111 fliftfuns 5631 dfmpo 6050 qliftfuns 6443 xpf1o 6667 caucvgprprlemaddq 7417 nfseq 10069 txcnp 12221 cnmpt1t 12235 cnmpt2t 12243 |
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