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Mirrors > Home > MPE Home > Th. List > axtgcont | Structured version Visualization version GIF version |
Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 26181. (Contributed by Thierry Arnoux, 16-Mar-2019.) |
Ref | Expression |
---|---|
axtrkg.p | ⊢ 𝑃 = (Base‘𝐺) |
axtrkg.d | ⊢ − = (dist‘𝐺) |
axtrkg.i | ⊢ 𝐼 = (Itv‘𝐺) |
axtrkg.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
axtgcont.1 | ⊢ (𝜑 → 𝑆 ⊆ 𝑃) |
axtgcont.2 | ⊢ (𝜑 → 𝑇 ⊆ 𝑃) |
axtgcont.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
axtgcont.4 | ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇) → 𝑢 ∈ (𝐴𝐼𝑣)) |
Ref | Expression |
---|---|
axtgcont | ⊢ (𝜑 → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axtgcont.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
2 | axtgcont.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇) → 𝑢 ∈ (𝐴𝐼𝑣)) | |
3 | 2 | 3expb 1112 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣)) |
4 | 3 | ralrimivva 3188 | . . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)) |
5 | simplr 765 | . . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢) | |
6 | simpll 763 | . . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑎 = 𝐴) | |
7 | simpr 485 | . . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣) | |
8 | 6, 7 | oveq12d 7163 | . . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣)) |
9 | 5, 8 | eleq12d 2904 | . . . . . 6 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣))) |
10 | 9 | cbvraldva 3457 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) → (∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣))) |
11 | 10 | cbvraldva 3457 | . . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣))) |
12 | 11 | rspcev 3620 | . . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)) → ∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦)) |
13 | 1, 4, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦)) |
14 | axtrkg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
15 | axtrkg.d | . . 3 ⊢ − = (dist‘𝐺) | |
16 | axtrkg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
17 | axtrkg.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
18 | axtgcont.1 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑃) | |
19 | axtgcont.2 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑃) | |
20 | 14, 15, 16, 17, 18, 19 | axtgcont1 26181 | . 2 ⊢ (𝜑 → (∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦))) |
21 | 13, 20 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑏 ∈ (𝑥𝐼𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 distcds 16562 TarskiGcstrkg 26143 Itvcitv 26149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-trkgb 26162 df-trkg 26166 |
This theorem is referenced by: f1otrg 26584 |
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