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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnutrd | Structured version Visualization version GIF version |
Description: Minimal universes are transitive. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
mnutrd.1 | ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} |
mnutrd.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) |
Ref | Expression |
---|---|
mnutrd | ⊢ (𝜑 → Tr 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnutrd.1 | . . . . 5 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} | |
2 | mnutrd.2 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑈 ∈ 𝑀) |
4 | simprr 772 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈) | |
5 | simprl 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑦) | |
6 | 1, 3, 4, 5 | mnutrcld 44250 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈) |
7 | 6 | ex 412 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈)) |
8 | 7 | alrimivv 1927 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈)) |
9 | dftr2 5285 | . 2 ⊢ (Tr 𝑈 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈)) | |
10 | 8, 9 | sylibr 234 | 1 ⊢ (𝜑 → Tr 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2108 {cab 2717 ∀wral 3067 ∃wrex 3076 ⊆ wss 3976 𝒫 cpw 4622 ∪ cuni 4931 Tr wtr 5283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-in 3983 df-ss 3993 df-pw 4624 df-sn 4649 df-uni 4932 df-tr 5284 |
This theorem is referenced by: mnugrud 44255 |
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