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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1309 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 31940. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1309.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
Ref | Expression |
---|---|
bnj1309 | ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1309.1 | . 2 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | |
2 | hbra1 3186 | . . . 4 ⊢ (∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 → ∀𝑥∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) | |
3 | 2 | bnj1352 31708 | . . 3 ⊢ ((𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) → ∀𝑥(𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)) |
4 | 3 | hbab 2785 | . 2 ⊢ (𝑤 ∈ {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} → ∀𝑥 𝑤 ∈ {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}) |
5 | 1, 4 | hbxfreq 2911 | 1 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1520 = wceq 1522 ∈ wcel 2080 {cab 2774 ∀wral 3104 ⊆ wss 3861 predc-bnj14 31567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-clab 2775 df-cleq 2787 df-clel 2862 df-ral 3109 |
This theorem is referenced by: bnj1311 31902 bnj1373 31908 bnj1498 31939 bnj1525 31947 bnj1523 31949 |
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