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Mirrors > Home > MPE Home > Th. List > en1unielOLD | Structured version Visualization version GIF version |
Description: Obsolete version of en1uniel 8683 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 8609 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 5590 | . . 3 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V) |
3 | uniexg 7506 | . . 3 ⊢ (𝑆 ∈ V → ∪ 𝑆 ∈ V) | |
4 | snidg 4561 | . . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆}) | |
5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ {∪ 𝑆}) |
6 | en1b 8678 | . . 3 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆}) | |
7 | 6 | biimpi 219 | . 2 ⊢ (𝑆 ≈ 1o → 𝑆 = {∪ 𝑆}) |
8 | 5, 7 | eleqtrrd 2834 | 1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3398 {csn 4527 ∪ cuni 4805 class class class wbr 5039 1oc1o 8173 ≈ cen 8601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-1o 8180 df-en 8605 |
This theorem is referenced by: (None) |
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