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Mirrors > Home > ILE Home > Th. List > op1stb | GIF version |
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.) |
Ref | Expression |
---|---|
op1stb.1 | ⊢ 𝐴 ∈ V |
op1stb.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
op1stb | ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op1stb.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
2 | op1stb.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | dfop 3751 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | inteqi 3822 | . . . 4 ⊢ ∩ 〈𝐴, 𝐵〉 = ∩ {{𝐴}, {𝐴, 𝐵}} |
5 | 1 | snex 4158 | . . . . . 6 ⊢ {𝐴} ∈ V |
6 | prexg 4183 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
7 | 1, 2, 6 | mp2an 423 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ V |
8 | 5, 7 | intpr 3850 | . . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}) |
9 | snsspr1 3715 | . . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
10 | df-ss 3124 | . . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}) | |
11 | 9, 10 | mpbi 144 | . . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴} |
12 | 8, 11 | eqtri 2185 | . . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴} |
13 | 4, 12 | eqtri 2185 | . . 3 ⊢ ∩ 〈𝐴, 𝐵〉 = {𝐴} |
14 | 13 | inteqi 3822 | . 2 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = ∩ {𝐴} |
15 | 1 | intsn 3853 | . 2 ⊢ ∩ {𝐴} = 𝐴 |
16 | 14, 15 | eqtri 2185 | 1 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∩ cin 3110 ⊆ wss 3111 {csn 3570 {cpr 3571 〈cop 3573 ∩ cint 3818 |
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-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
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-ral 2447 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-int 3819 |
This theorem is referenced by: elreldm 4824 op2ndb 5081 1stval2 6115 fundmen 6763 xpsnen 6778 |
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