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Mirrors > Home > MPE Home > Th. List > nfop | Structured version Visualization version 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 | dfopif 4755 | . 2 ⊢ 〈𝐴, 𝐵〉 = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | |
2 | nfop.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfel1 2915 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V |
4 | nfop.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfel1 2915 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V |
6 | 3, 5 | nfan 1906 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V) |
7 | 2 | nfsn 4598 | . . . 4 ⊢ Ⅎ𝑥{𝐴} |
8 | 2, 4 | nfpr 4581 | . . . 4 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
9 | 7, 8 | nfpr 4581 | . . 3 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}} |
10 | nfcv 2899 | . . 3 ⊢ Ⅎ𝑥∅ | |
11 | 6, 9, 10 | nfif 4444 | . 2 ⊢ Ⅎ𝑥if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) |
12 | 1, 11 | nfcxfr 2897 | 1 ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2114 Ⅎwnfc 2879 Vcvv 3398 ∅c0 4211 ifcif 4414 {csn 4516 {cpr 4518 〈cop 4522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-v 3400 df-dif 3846 df-un 3848 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 |
This theorem is referenced by: nfopd 4778 moop2 5359 iunopeqop 5378 fliftfuns 7080 dfmpo 7823 qliftfuns 8415 xpf1o 8729 nfseq 13470 txcnp 22371 cnmpt1t 22416 cnmpt2t 22424 flfcnp2 22758 bnj958 32491 bnj1000 32492 bnj1446 32596 bnj1447 32597 bnj1448 32598 bnj1466 32604 bnj1467 32605 bnj1519 32616 bnj1520 32617 bnj1529 32621 nosupbnd2 33562 noinfbnd2 33577 poimirlem26 35446 nfopdALT 36628 nfaov 44224 |
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