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Mirrors > Home > MPE Home > Th. List > tpid1g | Structured version Visualization version GIF version |
Description: Closed theorem form of tpid1 4706. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
tpid1g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix1i 1329 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷) |
3 | eltpg 4625 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) | |
4 | 2, 3 | mpbiri 260 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 {ctp 4573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-un 3943 df-sn 4570 df-pr 4572 df-tp 4574 |
This theorem is referenced by: tpnzd 4717 cplgr3v 27219 cyc3co2 30784 limsupequzlem 42010 |
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