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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 9872. (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 9780 | . . 3 ⊢ 0R ∈ R | |
2 | snssi 4280 | . . 3 ⊢ (0R ∈ R → {0R} ⊆ R) | |
3 | xpss2 5152 | . . 3 ⊢ ({0R} ⊆ R → (R × {0R}) ⊆ (R × R)) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ (R × {0R}) ⊆ (R × R) |
5 | df-r 9825 | . 2 ⊢ ℝ = (R × {0R}) | |
6 | df-c 9821 | . 2 ⊢ ℂ = (R × R) | |
7 | 4, 5, 6 | 3sstr4i 3607 | 1 ⊢ ℝ ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ⊆ wss 3540 {csn 4125 × cxp 5036 Rcnr 9566 0Rc0r 9567 ℂcc 9813 ℝcr 9814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-ni 9573 df-pli 9574 df-mi 9575 df-lti 9576 df-plpq 9609 df-mpq 9610 df-ltpq 9611 df-enq 9612 df-nq 9613 df-erq 9614 df-plq 9615 df-mq 9616 df-1nq 9617 df-rq 9618 df-ltnq 9619 df-np 9682 df-1p 9683 df-enr 9756 df-nr 9757 df-0r 9761 df-c 9821 df-r 9825 |
This theorem is referenced by: ax1cn 9849 bj-rrhatsscchat 32300 |
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