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Mirrors > Home > MPE Home > Th. List > rzal | Structured version Visualization version GIF version |
Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2809, ax-8 2108. (Revised by Gino Giotto, 2-Sep-2024.) |
Ref | Expression |
---|---|
rzal | ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2724 | . . . 4 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥})) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥})) |
3 | df-clab 2709 | . . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥) | |
4 | sbv 2091 | . . . . . 6 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥) | |
5 | 3, 4 | bitri 274 | . . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥) |
6 | 5 | bibi2i 337 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) |
7 | nbfal 1556 | . . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) | |
8 | pm2.21 123 | . . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) | |
9 | 7, 8 | sylbir 234 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ ⊥) → (𝑥 ∈ 𝐴 → 𝜑)) |
10 | 6, 9 | sylbi 216 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) → (𝑥 ∈ 𝐴 → 𝜑)) |
11 | 2, 10 | sylg 1825 | . 2 ⊢ (𝐴 = {𝑦 ∣ ⊥} → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
12 | dfnul4 4320 | . . 3 ⊢ ∅ = {𝑦 ∣ ⊥} | |
13 | 12 | eqeq2i 2744 | . 2 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥}) |
14 | df-ral 3061 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
15 | 11, 13, 14 | 3imtr4i 291 | 1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1539 = wceq 1541 ⊥wfal 1553 [wsb 2067 ∈ wcel 2106 {cab 2708 ∀wral 3060 ∅c0 4318 |
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-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-ral 3061 df-dif 3947 df-nul 4319 |
This theorem is referenced by: rexn0 4504 ral0 4506 ralf0 4507 ralidmOLD 4509 raaan 4514 raaanv 4515 raaan2 4518 iinrab2 5066 riinrab 5080 reusv2lem2 5390 cnvpo 6275 dffi3 9408 brdom3 10505 dedekind 11359 fimaxre2 12141 fiminre2 12144 mgm0 18557 sgrp0 18599 efgs1 19567 opnnei 22553 bddiblnc 25288 axcontlem12 28098 nbgr0edg 28479 prcliscplgr 28536 cplgr0v 28549 0vtxrgr 28698 0vconngr 29311 frgr1v 29389 ubthlem1 29986 rdgssun 36061 matunitlindf 36288 mbfresfi 36336 blbnd 36458 rrnequiv 36506 upbdrech2 43789 limsupubuz 44200 stoweidlem9 44496 fourierdlem31 44625 upwordnul 45365 upwordsing 45369 |
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