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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 4832 | . 2 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | |
2 | nfop.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfel1 2924 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V |
4 | nfop.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfel1 2924 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V |
6 | 3, 5 | nfan 1903 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V) |
7 | 2 | nfsn 4673 | . . . 4 ⊢ Ⅎ𝑥{𝐴} |
8 | 2, 4 | nfpr 4656 | . . . 4 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
9 | 7, 8 | nfpr 4656 | . . 3 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}} |
10 | nfcv 2908 | . . 3 ⊢ Ⅎ𝑥∅ | |
11 | 6, 9, 10 | nfif 4521 | . 2 ⊢ Ⅎ𝑥if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) |
12 | 1, 11 | nfcxfr 2906 | 1 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 Ⅎwnfc 2888 Vcvv 3448 ∅c0 4287 ifcif 4491 {csn 4591 {cpr 4593 ⟨cop 4597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 |
This theorem is referenced by: nfopd 4852 moop2 5464 iunopeqop 5483 fliftfuns 7264 dfmpo 8039 qliftfuns 8750 xpf1o 9090 nfseq 13923 txcnp 22987 cnmpt1t 23032 cnmpt2t 23040 flfcnp2 23374 nosupbnd2 27080 noinfbnd2 27095 bnj958 33592 bnj1000 33593 bnj1446 33697 bnj1447 33698 bnj1448 33699 bnj1466 33705 bnj1467 33706 bnj1519 33717 bnj1520 33718 bnj1529 33722 poimirlem26 36133 nfopdALT 37462 nfaov 45485 |
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