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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 3506 | . 2 ⊢ 〈𝐴, 𝐵〉 = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})} | |
2 | nfop.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfel1 2269 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V |
4 | nfop.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfel1 2269 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V |
6 | 2 | nfsn 3553 | . . . . . 6 ⊢ Ⅎ𝑥{𝐴} |
7 | 2, 4 | nfpr 3543 | . . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
8 | 6, 7 | nfpr 3543 | . . . . 5 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}} |
9 | 8 | nfcri 2252 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}} |
10 | 3, 5, 9 | nf3an 1530 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}) |
11 | 10 | nfab 2263 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})} |
12 | 1, 11 | nfcxfr 2255 | 1 ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 947 ∈ wcel 1465 {cab 2103 Ⅎwnfc 2245 Vcvv 2660 {csn 3497 {cpr 3498 〈cop 3500 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 |
This theorem is referenced by: nfopd 3692 moop2 4143 fliftfuns 5667 dfmpo 6088 qliftfuns 6481 xpf1o 6706 caucvgprprlemaddq 7484 nfseq 10196 txcnp 12367 cnmpt1t 12381 cnmpt2t 12389 |
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