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| Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6247 to extract the second member, op1sta 6245 for an alternate version, and op1st 8022 for the preferred version.) (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 4872 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} | 
| 4 | 3 | inteqi 4950 | . . . 4 ⊢ ∩ 〈𝐴, 𝐵〉 = ∩ {{𝐴}, {𝐴, 𝐵}} | 
| 5 | snex 5436 | . . . . . 6 ⊢ {𝐴} ∈ V | |
| 6 | prex 5437 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ V | |
| 7 | 5, 6 | intpr 4982 | . . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}) | 
| 8 | snsspr1 4814 | . . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
| 9 | dfss2 3969 | . . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}) | |
| 10 | 8, 9 | mpbi 230 | . . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴} | 
| 11 | 7, 10 | eqtri 2765 | . . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴} | 
| 12 | 4, 11 | eqtri 2765 | . . 3 ⊢ ∩ 〈𝐴, 𝐵〉 = {𝐴} | 
| 13 | 12 | inteqi 4950 | . 2 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = ∩ {𝐴} | 
| 14 | 1 | intsn 4984 | . 2 ⊢ ∩ {𝐴} = 𝐴 | 
| 15 | 13, 14 | eqtri 2765 | 1 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 {csn 4626 {cpr 4628 〈cop 4632 ∩ cint 4946 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-int 4947 | 
| This theorem is referenced by: elreldm 5946 op2ndb 6247 elxp5 7945 1stval2 8031 fundmen 9071 xpsnen 9095 | 
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