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| Mirrors > Home > MPE Home > Th. List > recnprss | Structured version Visualization version GIF version | ||
| Description: Both ℝ and ℂ are subsets of ℂ. (Contributed by Mario Carneiro, 10-Feb-2015.) |
| Ref | Expression |
|---|---|
| recnprss | ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elpri 4649 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
| 2 | ax-resscn 11212 | . . . 4 ⊢ ℝ ⊆ ℂ | |
| 3 | sseq1 4009 | . . . 4 ⊢ (𝑆 = ℝ → (𝑆 ⊆ ℂ ↔ ℝ ⊆ ℂ)) | |
| 4 | 2, 3 | mpbiri 258 | . . 3 ⊢ (𝑆 = ℝ → 𝑆 ⊆ ℂ) |
| 5 | eqimss 4042 | . . 3 ⊢ (𝑆 = ℂ → 𝑆 ⊆ ℂ) | |
| 6 | 4, 5 | jaoi 858 | . 2 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝑆 ⊆ ℂ) |
| 7 | 1, 6 | syl 17 | 1 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 {cpr 4628 ℂcc 11153 ℝcr 11154 |
| 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-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: dvres3 25948 dvres3a 25949 dvcnp 25954 dvnff 25959 dvnadd 25965 dvnres 25967 cpnord 25971 cpncn 25972 cpnres 25973 dvadd 25977 dvmul 25978 dvaddf 25979 dvmulf 25980 dvcmul 25981 dvcmulf 25982 dvco 25985 dvcof 25986 dvmptid 25995 dvmptc 25996 dvmptres2 26000 dvmptcmul 26002 dvmptfsum 26013 dvcnvlem 26014 dvcnv 26015 dvlip2 26034 taylfvallem1 26398 tayl0 26403 taylply2 26409 taylply2OLD 26410 taylply 26411 dvtaylp 26412 dvntaylp 26413 taylthlem1 26415 ulmdvlem1 26443 ulmdvlem3 26445 ulmdv 26446 dvsconst 44349 dvsid 44350 dvsef 44351 dvconstbi 44353 expgrowth 44354 dvdmsscn 45951 dvnmptdivc 45953 dvnmptconst 45956 dvnxpaek 45957 dvnmul 45958 dvnprodlem3 45963 |
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