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Mirrors > Home > MPE Home > Th. List > reltre | Structured version Visualization version GIF version |
Description: For all real numbers there is a smaller real number. (Contributed by AV, 5-Sep-2020.) |
Ref | Expression |
---|---|
reltre | ⊢ ∀𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑦 < 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2rem 11465 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ) | |
2 | breq1 5107 | . . . 4 ⊢ (𝑦 = (𝑥 − 1) → (𝑦 < 𝑥 ↔ (𝑥 − 1) < 𝑥)) | |
3 | 2 | adantl 482 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 = (𝑥 − 1)) → (𝑦 < 𝑥 ↔ (𝑥 − 1) < 𝑥)) |
4 | ltm1 11994 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝑥 − 1) < 𝑥) | |
5 | 1, 3, 4 | rspcedvd 3582 | . 2 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) |
6 | 5 | rgen 3065 | 1 ⊢ ∀𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑦 < 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∀wral 3063 ∃wrex 3072 class class class wbr 5104 (class class class)co 7354 ℝcr 11047 1c1 11049 < clt 11186 − cmin 11382 |
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 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-po 5544 df-so 5545 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 |
This theorem is referenced by: reltxrnmnf 13258 |
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