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Mirrors > Home > MPE Home > Th. List > unimax | Structured version Visualization version GIF version |
Description: Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.) |
Ref | Expression |
---|---|
unimax | ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 4018 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | sseq1 4021 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴)) | |
3 | 2 | elrab3 3696 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ 𝐴 ⊆ 𝐴)) |
4 | 1, 3 | mpbiri 258 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}) |
5 | sseq1 4021 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)) | |
6 | 5 | elrab 3695 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ 𝐴)) |
7 | 6 | simprbi 496 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝑦 ⊆ 𝐴) |
8 | 7 | rgen 3061 | . 2 ⊢ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴 |
9 | ssunieq 4948 | . . 3 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}) | |
10 | 9 | eqcomd 2741 | . 2 ⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴) |
11 | 4, 8, 10 | sylancl 586 | 1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 ∪ cuni 4912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-ss 3980 df-uni 4913 |
This theorem is referenced by: lssuni 20955 chsupid 31441 shatomistici 32390 lssats 38994 lpssat 38995 lssatle 38997 lssat 38998 mrelatglbALT 48785 mreclat 48786 toplatmeet 48792 |
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