![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dom0OLD | Structured version Visualization version GIF version |
Description: Obsolete version of dom0 9133 as of 29-Nov-2024. (Contributed by NM, 22-Nov-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dom0OLD | ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8976 | . . . . 5 ⊢ Rel ≼ | |
2 | 1 | brrelex1i 5738 | . . . 4 ⊢ (𝐴 ≼ ∅ → 𝐴 ∈ V) |
3 | 0domg 9131 | . . . 4 ⊢ (𝐴 ∈ V → ∅ ≼ 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ≼ ∅ → ∅ ≼ 𝐴) |
5 | 4 | pm4.71i 558 | . 2 ⊢ (𝐴 ≼ ∅ ↔ (𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴)) |
6 | sbthb 9125 | . 2 ⊢ ((𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴) ↔ 𝐴 ≈ ∅) | |
7 | en0 9044 | . 2 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
8 | 5, 6, 7 | 3bitri 296 | 1 ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∅c0 4326 class class class wbr 5152 ≈ cen 8967 ≼ cdom 8968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-er 8731 df-en 8971 df-dom 8972 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |