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Mirrors > Home > MPE Home > Th. List > tpid1g | Structured version Visualization version GIF version |
Description: Closed theorem form of tpid1 4729. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
tpid1g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix1i 1333 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷) |
3 | eltpg 4646 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) | |
4 | 2, 3 | mpbiri 257 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 {ctp 4590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3447 df-un 3915 df-sn 4587 df-pr 4589 df-tp 4591 |
This theorem is referenced by: tpnzd 4741 cplgr3v 28383 cyc3co2 31989 limsupequzlem 43953 |
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