![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11169. (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 11077 | . . 3 ⊢ 0R ∈ R | |
2 | snssi 4811 | . . 3 ⊢ (0R ∈ R → {0R} ⊆ R) | |
3 | xpss2 5696 | . . 3 ⊢ ({0R} ⊆ R → (R × {0R}) ⊆ (R × R)) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ (R × {0R}) ⊆ (R × R) |
5 | df-r 11122 | . 2 ⊢ ℝ = (R × {0R}) | |
6 | df-c 11118 | . 2 ⊢ ℂ = (R × R) | |
7 | 4, 5, 6 | 3sstr4i 4025 | 1 ⊢ ℝ ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ⊆ wss 3948 {csn 4628 × cxp 5674 Rcnr 10862 0Rc0r 10863 ℂcc 11110 ℝcr 11111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-omul 8473 df-er 8705 df-ec 8707 df-qs 8711 df-ni 10869 df-pli 10870 df-mi 10871 df-lti 10872 df-plpq 10905 df-mpq 10906 df-ltpq 10907 df-enq 10908 df-nq 10909 df-erq 10910 df-plq 10911 df-mq 10912 df-1nq 10913 df-rq 10914 df-ltnq 10915 df-np 10978 df-1p 10979 df-enr 11052 df-nr 11053 df-0r 11057 df-c 11118 df-r 11122 |
This theorem is referenced by: ax1cn 11146 bj-rrhatsscchat 36203 |
Copyright terms: Public domain | W3C validator |