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| Mirrors > Home > MPE Home > Th. List > domwdom | Structured version Visualization version GIF version | ||
| Description: Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
| Ref | Expression |
|---|---|
| domwdom | ⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neqne 2948 | . . . . . . 7 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
| 2 | 1 | adantl 481 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
| 3 | reldom 8991 | . . . . . . . . 9 ⊢ Rel ≼ | |
| 4 | 3 | brrelex1i 5741 | . . . . . . . 8 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ∈ V) |
| 5 | 0sdomg 9144 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | |
| 6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝑋 ≼ 𝑌 → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
| 7 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
| 8 | 2, 7 | mpbird 257 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∅ ≺ 𝑋) |
| 9 | simpl 482 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≼ 𝑌) | |
| 10 | fodomr 9168 | . . . . 5 ⊢ ((∅ ≺ 𝑋 ∧ 𝑋 ≼ 𝑌) → ∃𝑦 𝑦:𝑌–onto→𝑋) | |
| 11 | 8, 9, 10 | syl2anc 584 | . . . 4 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∃𝑦 𝑦:𝑌–onto→𝑋) |
| 12 | 11 | ex 412 | . . 3 ⊢ (𝑋 ≼ 𝑌 → (¬ 𝑋 = ∅ → ∃𝑦 𝑦:𝑌–onto→𝑋)) |
| 13 | 12 | orrd 864 | . 2 ⊢ (𝑋 ≼ 𝑌 → (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋)) |
| 14 | 3 | brrelex2i 5742 | . . 3 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V) |
| 15 | brwdom 9607 | . . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋))) | |
| 16 | 14, 15 | syl 17 | . 2 ⊢ (𝑋 ≼ 𝑌 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋))) |
| 17 | 13, 16 | mpbird 257 | 1 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 class class class wbr 5143 –onto→wfo 6559 ≼ cdom 8983 ≺ csdm 8984 ≼* cwdom 9604 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 df-dom 8987 df-sdom 8988 df-wdom 9605 |
| This theorem is referenced by: wdomen1 9616 wdomen2 9617 wdom2d 9620 wdomima2g 9626 unxpwdom2 9628 unxpwdom 9629 harwdom 9631 wdomfil 10101 wdomnumr 10104 pwdjudom 10255 hsmexlem1 10466 hsmexlem4 10469 |
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