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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 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝑈 ∈ 𝑀) |
4 | mnuprd.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
5 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ∈ 𝑈) |
6 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅) | |
7 | 0ss 4343 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
8 | 6, 7 | eqsstrdi 4014 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 ⊆ 𝐵) |
9 | ssidd 3983 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐵 ⊆ 𝐵) | |
10 | 1, 3, 5, 8, 9 | mnuprssd 40680 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈) |
11 | eqid 2820 | . . 3 ⊢ {{∅, {𝐴}}, {{∅}, {𝐵}}} = {{∅, {𝐴}}, {{∅}, {𝐵}}} | |
12 | 2 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑈 ∈ 𝑀) |
13 | mnuprd.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
14 | 13 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ∈ 𝑈) |
15 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐵 ∈ 𝑈) |
16 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅) | |
17 | 1, 11, 12, 14, 15, 16 | mnuprdlem4 40686 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈) |
18 | 10, 17 | pm2.61dan 811 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1534 = wceq 1536 ∈ wcel 2113 {cab 2798 ∀wral 3137 ∃wrex 3138 ⊆ wss 3929 ∅c0 4284 𝒫 cpw 4532 {csn 4560 {cpr 4562 ∪ cuni 4831 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-pw 4534 df-sn 4561 df-pr 4563 df-uni 4832 |
This theorem is referenced by: mnuund 40689 mnurndlem2 40693 mnugrud 40695 |
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