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Mirrors > Home > ILE Home > Th. List > djudoml | GIF version |
Description: A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
Ref | Expression |
---|---|
djudoml | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ (𝐴 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 7040 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 1 | funmpt2 5251 | . . . 4 ⊢ Fun inl |
3 | simpl 109 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
4 | resfunexg 5733 | . . . 4 ⊢ ((Fun inl ∧ 𝐴 ∈ 𝑉) → (inl ↾ 𝐴) ∈ V) | |
5 | 2, 3, 4 | sylancr 414 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl ↾ 𝐴) ∈ V) |
6 | inlresf1 7054 | . . 3 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) | |
7 | f1eq1 5412 | . . . 4 ⊢ (𝑓 = (inl ↾ 𝐴) → (𝑓:𝐴–1-1→(𝐴 ⊔ 𝐵) ↔ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵))) | |
8 | 7 | spcegv 2825 | . . 3 ⊢ ((inl ↾ 𝐴) ∈ V → ((inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) → ∃𝑓 𝑓:𝐴–1-1→(𝐴 ⊔ 𝐵))) |
9 | 5, 6, 8 | mpisyl 1446 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑓 𝑓:𝐴–1-1→(𝐴 ⊔ 𝐵)) |
10 | djuex 7036 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) | |
11 | brdomg 6742 | . . 3 ⊢ ((𝐴 ⊔ 𝐵) ∈ V → (𝐴 ≼ (𝐴 ⊔ 𝐵) ↔ ∃𝑓 𝑓:𝐴–1-1→(𝐴 ⊔ 𝐵))) | |
12 | 10, 11 | syl 14 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ (𝐴 ⊔ 𝐵) ↔ ∃𝑓 𝑓:𝐴–1-1→(𝐴 ⊔ 𝐵))) |
13 | 9, 12 | mpbird 167 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 ∅c0 3422 〈cop 3594 class class class wbr 4000 ↾ cres 4625 Fun wfun 5206 –1-1→wf1 5209 ≼ cdom 6733 ⊔ cdju 7030 inlcinl 7038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-suc 4368 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-1st 6135 df-2nd 6136 df-1o 6411 df-dom 6736 df-dju 7031 df-inl 7040 |
This theorem is referenced by: (None) |
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