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| Mirrors > Home > MPE Home > Th. List > prssg | Structured version Visualization version GIF version | ||
| Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
| Ref | Expression |
|---|---|
| prssg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snssg 4783 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶)) | |
| 2 | snssg 4783 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ 𝐶 ↔ {𝐵} ⊆ 𝐶)) | |
| 3 | 1, 2 | bi2anan9 638 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶))) |
| 4 | unss 4190 | . . 3 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶) | |
| 5 | df-pr 4629 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 6 | 5 | sseq1i 4012 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶) |
| 7 | 4, 6 | bitr4i 278 | . 2 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶) |
| 8 | 3, 7 | bitrdi 287 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: prss 4820 prssi 4821 prsspwg 4823 ssprss 4824 prelpw 5451 hashdmpropge2 14522 lspvadd 21095 umgredgprv 29124 usgredgprvALT 29212 dfnbgr2 29354 nbuhgr 29360 uhgrnbgr0nb 29371 2wlkdlem6 29951 1wlkdlem2 30157 coss0 38480 dihmeetlem2N 41301 mnuprdlem2 44292 fourierdlem20 46142 fourierdlem50 46171 fourierdlem54 46175 fourierdlem64 46185 fourierdlem76 46197 omeunle 46531 dfclnbgr2 47810 dfsclnbgr2 47832 dfvopnbgr2 47839 isubgr3stgrlem7 47939 |
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