Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > opelreal | Structured version Visualization version GIF version |
Description: Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
opelreal | ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . 2 ⊢ 0R = 0R | |
2 | df-r 9946 | . . . 4 ⊢ ℝ = (R × {0R}) | |
3 | 2 | eleq2i 2693 | . . 3 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 〈𝐴, 0R〉 ∈ (R × {0R})) |
4 | opelxp 5146 | . . 3 ⊢ (〈𝐴, 0R〉 ∈ (R × {0R}) ↔ (𝐴 ∈ R ∧ 0R ∈ {0R})) | |
5 | 0r 9901 | . . . . . 6 ⊢ 0R ∈ R | |
6 | 5 | elexi 3213 | . . . . 5 ⊢ 0R ∈ V |
7 | 6 | elsn 4192 | . . . 4 ⊢ (0R ∈ {0R} ↔ 0R = 0R) |
8 | 7 | anbi2i 730 | . . 3 ⊢ ((𝐴 ∈ R ∧ 0R ∈ {0R}) ↔ (𝐴 ∈ R ∧ 0R = 0R)) |
9 | 3, 4, 8 | 3bitri 286 | . 2 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ (𝐴 ∈ R ∧ 0R = 0R)) |
10 | 1, 9 | mpbiran2 954 | 1 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 〈cop 4183 × cxp 5112 Rcnr 9687 0Rc0r 9688 ℝcr 9935 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-mpq 9731 df-ltpq 9732 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-mq 9737 df-1nq 9738 df-rq 9739 df-ltnq 9740 df-np 9803 df-1p 9804 df-enr 9877 df-nr 9878 df-0r 9882 df-r 9946 |
This theorem is referenced by: ltresr 9961 ax1cn 9970 axaddrcl 9973 axmulrcl 9975 axrnegex 9983 axrrecex 9984 axcnre 9985 axpre-sup 9990 |
Copyright terms: Public domain | W3C validator |