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Mirrors > Home > MPE Home > Th. List > axresscn | Structured version Visualization version GIF version |
Description: The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 11115. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axresscn | ⊢ ℝ ⊆ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 11023 | . . 3 ⊢ 0R ∈ R | |
2 | snssi 4773 | . . 3 ⊢ (0R ∈ R → {0R} ⊆ R) | |
3 | xpss2 5658 | . . 3 ⊢ ({0R} ⊆ R → (R × {0R}) ⊆ (R × R)) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ (R × {0R}) ⊆ (R × R) |
5 | df-r 11068 | . 2 ⊢ ℝ = (R × {0R}) | |
6 | df-c 11064 | . 2 ⊢ ℂ = (R × R) | |
7 | 4, 5, 6 | 3sstr4i 3992 | 1 ⊢ ℝ ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ⊆ wss 3915 {csn 4591 × cxp 5636 Rcnr 10808 0Rc0r 10809 ℂcc 11056 ℝcr 11057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-ec 8657 df-qs 8661 df-ni 10815 df-pli 10816 df-mi 10817 df-lti 10818 df-plpq 10851 df-mpq 10852 df-ltpq 10853 df-enq 10854 df-nq 10855 df-erq 10856 df-plq 10857 df-mq 10858 df-1nq 10859 df-rq 10860 df-ltnq 10861 df-np 10924 df-1p 10925 df-enr 10998 df-nr 10999 df-0r 11003 df-c 11064 df-r 11068 |
This theorem is referenced by: ax1cn 11092 bj-rrhatsscchat 35736 |
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