Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4399 | . 2 ⊢ 〈𝐴, 𝐵〉 = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | |
2 | nfop.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfel1 2779 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V |
4 | nfop.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfel1 2779 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V |
6 | 3, 5 | nfan 1828 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V) |
7 | 2 | nfsn 4242 | . . . 4 ⊢ Ⅎ𝑥{𝐴} |
8 | 2, 4 | nfpr 4232 | . . . 4 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
9 | 7, 8 | nfpr 4232 | . . 3 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}} |
10 | nfcv 2764 | . . 3 ⊢ Ⅎ𝑥∅ | |
11 | 6, 9, 10 | nfif 4115 | . 2 ⊢ Ⅎ𝑥if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) |
12 | 1, 11 | nfcxfr 2762 | 1 ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∈ wcel 1990 Ⅎwnfc 2751 Vcvv 3200 ∅c0 3915 ifcif 4086 {csn 4177 {cpr 4179 〈cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: nfopd 4419 moop2 4966 iunopeqop 4981 fliftfuns 6564 dfmpt2 7267 qliftfuns 7834 xpf1o 8122 nfseq 12811 txcnp 21423 cnmpt1t 21468 cnmpt2t 21476 flfcnp2 21811 bnj958 31010 bnj1000 31011 bnj1446 31113 bnj1447 31114 bnj1448 31115 bnj1466 31121 bnj1467 31122 bnj1519 31133 bnj1520 31134 bnj1529 31138 nosupbnd2 31862 poimirlem26 33435 nfopdALT 34258 nfaov 41259 |
Copyright terms: Public domain | W3C validator |