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Mirrors > Home > MPE Home > Th. List > en1unielOLD | Structured version Visualization version GIF version |
Description: Obsolete version of en1uniel 8793 as of 24-Sep-2024. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
en1unielOLD | ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 8713 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 5643 | . . 3 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V) |
3 | uniexg 7585 | . . 3 ⊢ (𝑆 ∈ V → ∪ 𝑆 ∈ V) | |
4 | snidg 4601 | . . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆}) | |
5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ {∪ 𝑆}) |
6 | en1b 8788 | . . 3 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆}) | |
7 | 6 | biimpi 215 | . 2 ⊢ (𝑆 ≈ 1o → 𝑆 = {∪ 𝑆}) |
8 | 5, 7 | eleqtrrd 2844 | 1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 Vcvv 3431 {csn 4567 ∪ cuni 4845 class class class wbr 5079 1oc1o 8275 ≈ cen 8705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-1o 8282 df-en 8709 |
This theorem is referenced by: (None) |
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