| Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > mnuprd | Structured version Visualization version GIF version | ||
| Description: Minimal universes are closed under pairing. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
| Ref | Expression |
|---|---|
| mnuprd.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))))} |
| mnuprd.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) |
| mnuprd.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
| mnuprd.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
| Ref | Expression |
|---|---|
| mnuprd | ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mnuprd.1 | . . 3 ⊢ 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 | mnuprd.2 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
| 3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝑈 ∈ 𝑀) |
| 4 | mnuprd.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
| 5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ∈ 𝑈) |
| 6 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅) | |
| 7 | 0ss 4400 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
| 8 | 6, 7 | eqsstrdi 4028 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 ⊆ 𝐵) |
| 9 | ssidd 4007 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ⊆ 𝐵) | |
| 10 | 1, 3, 5, 8, 9 | mnuprssd 44288 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈) |
| 11 | eqid 2737 | . . 3 ⊢ {{∅, {𝐴}}, {{∅}, {𝐵}}} = {{∅, {𝐴}}, {{∅}, {𝐵}}} | |
| 12 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑈 ∈ 𝑀) |
| 13 | mnuprd.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 14 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ 𝑈) |
| 15 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐵 ∈ 𝑈) |
| 16 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅) | |
| 17 | 1, 11, 12, 14, 15, 16 | mnuprdlem4 44294 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈) |
| 18 | 10, 17 | pm2.61dan 813 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {cpr 4628 ∪ cuni 4907 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 |
| This theorem is referenced by: mnuund 44297 mnurndlem2 44301 mnugrud 44303 |
| Copyright terms: Public domain | W3C validator |