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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 25780. (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 1155 | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣)) |
4 | 3 | ralrimivva 3180 | . . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)) |
5 | simplr 787 | . . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢) | |
6 | simpll 785 | . . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑎 = 𝐴) | |
7 | simpr 479 | . . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣) | |
8 | 6, 7 | oveq12d 6923 | . . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣)) |
9 | 5, 8 | eleq12d 2900 | . . . . . 6 ⊢ (((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣))) |
10 | 9 | cbvraldva 3389 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑥 = 𝑢) → (∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣))) |
11 | 10 | cbvraldva 3389 | . . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣))) |
12 | 11 | rspcev 3526 | . . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑇 𝑢 ∈ (𝐴𝐼𝑣)) → ∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 𝑥 ∈ (𝑎𝐼𝑦)) |
13 | 1, 4, 12 | syl2anc 581 | . 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 25780 | . 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 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ∃wrex 3118 ⊆ wss 3798 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 distcds 16314 TarskiGcstrkg 25742 Itvcitv 25748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-trkgb 25761 df-trkg 25765 |
This theorem is referenced by: f1otrg 26170 |
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