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Mirrors > Home > MPE Home > Th. List > prssOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of prss 4291 as of 23-Jul-2021. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
prss.1 | ⊢ 𝐴 ∈ V |
prss.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
prssOLD | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unss 3749 | . 2 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶) | |
2 | prss.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 2 | snss 4259 | . . 3 ⊢ (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶) |
4 | prss.2 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | 4 | snss 4259 | . . 3 ⊢ (𝐵 ∈ 𝐶 ↔ {𝐵} ⊆ 𝐶) |
6 | 3, 5 | anbi12i 729 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶)) |
7 | df-pr 4128 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
8 | 7 | sseq1i 3592 | . 2 ⊢ ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶) |
9 | 1, 6, 8 | 3bitr4i 291 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 {csn 4125 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 |
This theorem is referenced by: (None) |
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