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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 25167. (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 1258 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣)) |
4 | 3 | ralrimivva 2954 | . . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)) |
5 | simplr 788 | . . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢) | |
6 | simpll 786 | . . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑎 = 𝐴) | |
7 | simpr 476 | . . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣) | |
8 | 6, 7 | oveq12d 6567 | . . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣)) |
9 | 5, 8 | eleq12d 2682 | . . . . . 6 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣))) |
10 | 9 | cbvraldva 3153 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) → (∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣))) |
11 | 10 | cbvraldva 3153 | . . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣))) |
12 | 11 | rspcev 3282 | . . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)) → ∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦)) |
13 | 1, 4, 12 | syl2anc 691 | . 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 25167 | . 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-trkgb 25148 df-trkg 25152 |
This theorem is referenced by: f1otrg 25551 |
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