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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 3670 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | inteqi 3741 | . . . 4 ⊢ ∩ 〈𝐴, 𝐵〉 = ∩ {{𝐴}, {𝐴, 𝐵}} |
5 | 1 | snex 4069 | . . . . . 6 ⊢ {𝐴} ∈ V |
6 | prexg 4093 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
7 | 1, 2, 6 | mp2an 420 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ V |
8 | 5, 7 | intpr 3769 | . . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}) |
9 | snsspr1 3634 | . . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
10 | df-ss 3050 | . . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}) | |
11 | 9, 10 | mpbi 144 | . . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴} |
12 | 8, 11 | eqtri 2135 | . . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴} |
13 | 4, 12 | eqtri 2135 | . . 3 ⊢ ∩ 〈𝐴, 𝐵〉 = {𝐴} |
14 | 13 | inteqi 3741 | . 2 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = ∩ {𝐴} |
15 | 1 | intsn 3772 | . 2 ⊢ ∩ {𝐴} = 𝐴 |
16 | 14, 15 | eqtri 2135 | 1 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1314 ∈ wcel 1463 Vcvv 2657 ∩ cin 3036 ⊆ wss 3037 {csn 3493 {cpr 3494 〈cop 3496 ∩ cint 3737 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-int 3738 |
This theorem is referenced by: elreldm 4725 op2ndb 4980 1stval2 6007 fundmen 6654 xpsnen 6668 |
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