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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 3579 | . 2 ⊢ 〈𝐴, 𝐵〉 = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
2 | nfop.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfel1 2317 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V |
4 | nfop.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfel1 2317 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V |
6 | 2 | nfsn 3630 | . . . . . 6 ⊢ Ⅎ𝑥{𝐴} |
7 | 2, 4 | nfpr 3620 | . . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
8 | 6, 7 | nfpr 3620 | . . . . 5 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}} |
9 | 8 | nfcri 2300 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}} |
10 | 3, 5, 9 | nf3an 1553 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}) |
11 | 10 | nfab 2311 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})} |
12 | 1, 11 | nfcxfr 2303 | 1 ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 967 ∈ wcel 2135 {cab 2150 Ⅎwnfc 2293 Vcvv 2721 {csn 3570 {cpr 3571 〈cop 3573 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 |
This theorem is referenced by: nfopd 3769 moop2 4223 fliftfuns 5760 dfmpo 6182 qliftfuns 6576 xpf1o 6801 caucvgprprlemaddq 7640 nfseq 10380 txcnp 12812 cnmpt1t 12826 cnmpt2t 12834 |
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