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Mirrors > Home > MPE Home > Th. List > r19.2uz | Structured version Visualization version GIF version |
Description: A version of r19.2z 4251 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.) |
Ref | Expression |
---|---|
rexuz3.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
r19.2uz | ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∃𝑘 ∈ 𝑍 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 11936 | . . . . . 6 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ) | |
2 | uzid 11941 | . . . . . 6 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗)) | |
3 | ne0i 4119 | . . . . . 6 ⊢ (𝑗 ∈ (ℤ≥‘𝑗) → (ℤ≥‘𝑗) ≠ ∅) | |
4 | 1, 2, 3 | 3syl 18 | . . . . 5 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑗) ≠ ∅) |
5 | rexuz3.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | 4, 5 | eleq2s 2894 | . . . 4 ⊢ (𝑗 ∈ 𝑍 → (ℤ≥‘𝑗) ≠ ∅) |
7 | r19.2z 4251 | . . . 4 ⊢ (((ℤ≥‘𝑗) ≠ ∅ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ∃𝑘 ∈ (ℤ≥‘𝑗)𝜑) | |
8 | 6, 7 | sylan 576 | . . 3 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ∃𝑘 ∈ (ℤ≥‘𝑗)𝜑) |
9 | 5 | uztrn2 11944 | . . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
10 | 9 | ex 402 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → (𝑘 ∈ (ℤ≥‘𝑗) → 𝑘 ∈ 𝑍)) |
11 | 10 | anim1d 605 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → ((𝑘 ∈ (ℤ≥‘𝑗) ∧ 𝜑) → (𝑘 ∈ 𝑍 ∧ 𝜑))) |
12 | 11 | reximdv2 3192 | . . . 4 ⊢ (𝑗 ∈ 𝑍 → (∃𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∃𝑘 ∈ 𝑍 𝜑)) |
13 | 12 | imp 396 | . . 3 ⊢ ((𝑗 ∈ 𝑍 ∧ ∃𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ∃𝑘 ∈ 𝑍 𝜑) |
14 | 8, 13 | syldan 586 | . 2 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ∃𝑘 ∈ 𝑍 𝜑) |
15 | 14 | rexlimiva 3207 | 1 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∃𝑘 ∈ 𝑍 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ∀wral 3087 ∃wrex 3088 ∅c0 4113 ‘cfv 6099 ℤcz 11662 ℤ≥cuz 11926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-pre-lttri 10296 ax-pre-lttrn 10297 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-ov 6879 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-neg 10557 df-z 11663 df-uz 11927 |
This theorem is referenced by: lmcls 21432 1stccnp 21591 iscmet3lem1 23414 iscmet3lem2 23415 uniioombllem6 23693 ulmcau 24487 ulmbdd 24490 ulmcn 24491 ulmdvlem3 24494 iblulm 24499 |
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