MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dif1en Structured version   Visualization version   GIF version

Theorem dif1en 9200
Description: If a set 𝐴 is equinumerous to the successor of an ordinal 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals. (Revised by BTernaryTau, 6-Jan-2025.)
Assertion
Ref Expression
dif1en ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1en
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 simp1 1137 . . 3 ((𝐴 ≈ suc 𝑀𝑋𝐴𝑀 ∈ On) → 𝐴 ≈ suc 𝑀)
2 encv 8993 . . . . 5 (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V))
32simpld 494 . . . 4 (𝐴 ≈ suc 𝑀𝐴 ∈ V)
433anim1i 1153 . . 3 ((𝐴 ≈ suc 𝑀𝑋𝐴𝑀 ∈ On) → (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On))
5 bren 8995 . . . 4 (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀)
6 sucidg 6465 . . . . . . . . . . . . 13 (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
7 f1ocnvdm 7305 . . . . . . . . . . . . . . . 16 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
873adant2 1132 . . . . . . . . . . . . . . 15 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
9 f1ofvswap 7326 . . . . . . . . . . . . . . 15 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴 ∧ (𝑓𝑀) ∈ 𝐴) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
108, 9syld3an3 1411 . . . . . . . . . . . . . 14 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
11 f1ocnvfv2 7297 . . . . . . . . . . . . . . . . . . 19 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (𝑓‘(𝑓𝑀)) = 𝑀)
1211opeq2d 4880 . . . . . . . . . . . . . . . . . 18 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → ⟨𝑋, (𝑓‘(𝑓𝑀))⟩ = ⟨𝑋, 𝑀⟩)
1312preq1d 4739 . . . . . . . . . . . . . . . . 17 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩} = {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})
1413uneq2d 4168 . . . . . . . . . . . . . . . 16 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) = ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}))
1514f1oeq1d 6843 . . . . . . . . . . . . . . 15 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀))
16153adant2 1132 . . . . . . . . . . . . . 14 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀))
1710, 16mpbid 232 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
186, 17syl3an3 1166 . . . . . . . . . . . 12 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
19183adant3r1 1183 . . . . . . . . . . 11 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
20 f1ofun 6850 . . . . . . . . . . 11 (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀 → Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}))
21 opex 5469 . . . . . . . . . . . . . 14 𝑋, 𝑀⟩ ∈ V
2221prid1 4762 . . . . . . . . . . . . 13 𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}
23 elun2 4183 . . . . . . . . . . . . 13 (⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩} → ⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}))
2422, 23ax-mp 5 . . . . . . . . . . . 12 𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})
25 funopfv 6958 . . . . . . . . . . . 12 (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) → (⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀))
2624, 25mpi 20 . . . . . . . . . . 11 (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀)
2719, 20, 263syl 18 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀)
28 simpr2 1196 . . . . . . . . . . 11 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → 𝑋𝐴)
29 f1ocnvfv 7298 . . . . . . . . . . 11 ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀𝑋𝐴) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀 → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋))
3019, 28, 29syl2anc 584 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀 → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋))
3127, 30mpd 15 . . . . . . . . 9 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋)
3231sneqd 4638 . . . . . . . 8 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)} = {𝑋})
3332difeq2d 4126 . . . . . . 7 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) = (𝐴 ∖ {𝑋}))
34 simpr1 1195 . . . . . . . . 9 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → 𝐴 ∈ V)
35 3simpc 1151 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On) → (𝑋𝐴𝑀 ∈ On))
3635anim2i 617 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝑋𝐴𝑀 ∈ On)))
37 3anass 1095 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On) ↔ (𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝑋𝐴𝑀 ∈ On)))
3836, 37sylibr 234 . . . . . . . . 9 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On))
3934, 38jca 511 . . . . . . . 8 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)))
40 simpl 482 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → 𝐴 ∈ V)
41 simpr3 1197 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → 𝑀 ∈ On)
4240, 41jca 511 . . . . . . . . 9 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → (𝐴 ∈ V ∧ 𝑀 ∈ On))
43 simpr 484 . . . . . . . . 9 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On))
4442, 43jca 511 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → ((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)))
45 vex 3484 . . . . . . . . . . . 12 𝑓 ∈ V
4645resex 6047 . . . . . . . . . . 11 (𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∈ V
47 prex 5437 . . . . . . . . . . 11 {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩} ∈ V
4846, 47unex 7764 . . . . . . . . . 10 ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) ∈ V
49 dif1enlem 9196 . . . . . . . . . 10 (((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) ∈ V ∧ 𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
5048, 49mp3anl1 1457 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
5118, 50sylan2 593 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
5239, 44, 513syl 18 . . . . . . 7 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
5333, 52eqbrtrrd 5167 . . . . . 6 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
5453ex 412 . . . . 5 (𝑓:𝐴1-1-onto→suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5554exlimiv 1930 . . . 4 (∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
565, 55sylbi 217 . . 3 (𝐴 ≈ suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
571, 4, 56sylc 65 . 2 ((𝐴 ≈ suc 𝑀𝑋𝐴𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
58573comr 1126 1 ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cdif 3948  cun 3949  {csn 4626  {cpr 4628  cop 4632   class class class wbr 5143  ccnv 5684  cres 5687  Oncon0 6384  suc csuc 6386  Fun wfun 6555  1-1-ontowf1o 6560  cfv 6561  cen 8982
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-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-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  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-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-en 8986
This theorem is referenced by:  dif1ennn  9201
  Copyright terms: Public domain W3C validator