MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pospo Structured version   Visualization version   GIF version
Theorem pospo 18298
Description: Write a poset structure in terms of the proper-class poset predicate (strict less than version). (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pospo.b 𝐡 = (Baseβ€˜πΎ)
pospo.l ≀ = (leβ€˜πΎ)
pospo.s < = (ltβ€˜πΎ)
Assertion
Ref Expression
pospo (𝐾 ∈ 𝑉 β†’ (𝐾 ∈ Poset ↔ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )))
Proof of Theorem pospo
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pospo.s . . . . 5 < = (ltβ€˜πΎ)
21pltirr 18288 . . . 4 ((𝐾 ∈ Poset ∧ π‘₯ ∈ 𝐡) β†’ Β¬ π‘₯ < π‘₯)
3 pospo.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
43, 1plttr 18295 . . . 4 ((𝐾 ∈ Poset ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ < 𝑦 ∧ 𝑦 < 𝑧) β†’ π‘₯ < 𝑧))
52, 4ispod 5598 . . 3 (𝐾 ∈ Poset β†’ < Po 𝐡)
6 relres 6011 . . . . 5 Rel ( I β†Ύ 𝐡)
76a1i 11 . . . 4 (𝐾 ∈ Poset β†’ Rel ( I β†Ύ 𝐡))
8 opabresid 6050 . . . . . . . 8 ( I β†Ύ 𝐡) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 = π‘₯)}
98eqcomi 2742 . . . . . . 7 {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 = π‘₯)} = ( I β†Ύ 𝐡)
109eleq2i 2826 . . . . . 6 (⟨π‘₯, π‘¦βŸ© ∈ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 = π‘₯)} ↔ ⟨π‘₯, π‘¦βŸ© ∈ ( I β†Ύ 𝐡))
11 opabidw 5525 . . . . . 6 (⟨π‘₯, π‘¦βŸ© ∈ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ 𝑦 = π‘₯)} ↔ (π‘₯ ∈ 𝐡 ∧ 𝑦 = π‘₯))
1210, 11bitr3i 277 . . . . 5 (⟨π‘₯, π‘¦βŸ© ∈ ( I β†Ύ 𝐡) ↔ (π‘₯ ∈ 𝐡 ∧ 𝑦 = π‘₯))
13 pospo.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
143, 13posref 18271 . . . . . . 7 ((𝐾 ∈ Poset ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ≀ π‘₯)
15 df-br 5150 . . . . . . . 8 (π‘₯ ≀ 𝑦 ↔ ⟨π‘₯, π‘¦βŸ© ∈ ≀ )
16 breq2 5153 . . . . . . . 8 (𝑦 = π‘₯ β†’ (π‘₯ ≀ 𝑦 ↔ π‘₯ ≀ π‘₯))
1715, 16bitr3id 285 . . . . . . 7 (𝑦 = π‘₯ β†’ (⟨π‘₯, π‘¦βŸ© ∈ ≀ ↔ π‘₯ ≀ π‘₯))
1814, 17syl5ibrcom 246 . . . . . 6 ((𝐾 ∈ Poset ∧ π‘₯ ∈ 𝐡) β†’ (𝑦 = π‘₯ β†’ ⟨π‘₯, π‘¦βŸ© ∈ ≀ ))
1918expimpd 455 . . . . 5 (𝐾 ∈ Poset β†’ ((π‘₯ ∈ 𝐡 ∧ 𝑦 = π‘₯) β†’ ⟨π‘₯, π‘¦βŸ© ∈ ≀ ))
2012, 19biimtrid 241 . . . 4 (𝐾 ∈ Poset β†’ (⟨π‘₯, π‘¦βŸ© ∈ ( I β†Ύ 𝐡) β†’ ⟨π‘₯, π‘¦βŸ© ∈ ≀ ))
217, 20relssdv 5789 . . 3 (𝐾 ∈ Poset β†’ ( I β†Ύ 𝐡) βŠ† ≀ )
225, 21jca 513 . 2 (𝐾 ∈ Poset β†’ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ ))
23 simpl 484 . . . 4 ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) β†’ 𝐾 ∈ 𝑉)
243a1i 11 . . . 4 ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) β†’ 𝐡 = (Baseβ€˜πΎ))
2513a1i 11 . . . 4 ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) β†’ ≀ = (leβ€˜πΎ))
26 equid 2016 . . . . . 6 π‘₯ = π‘₯
27 simpr 486 . . . . . . 7 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ 𝐡)
28 resieq 5993 . . . . . . 7 ((π‘₯ ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯( I β†Ύ 𝐡)π‘₯ ↔ π‘₯ = π‘₯))
2927, 27, 28syl2anc 585 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯( I β†Ύ 𝐡)π‘₯ ↔ π‘₯ = π‘₯))
3026, 29mpbiri 258 . . . . 5 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡) β†’ π‘₯( I β†Ύ 𝐡)π‘₯)
31 simplrr 777 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡) β†’ ( I β†Ύ 𝐡) βŠ† ≀ )
3231ssbrd 5192 . . . . 5 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯( I β†Ύ 𝐡)π‘₯ β†’ π‘₯ ≀ π‘₯))
3330, 32mpd 15 . . . 4 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ≀ π‘₯)
343, 13, 1pleval2i 18289 . . . . . 6 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ ≀ 𝑦 β†’ (π‘₯ < 𝑦 ∨ π‘₯ = 𝑦)))
35343adant1 1131 . . . . 5 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ ≀ 𝑦 β†’ (π‘₯ < 𝑦 ∨ π‘₯ = 𝑦)))
363, 13, 1pleval2i 18289 . . . . . . 7 ((𝑦 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) β†’ (𝑦 ≀ π‘₯ β†’ (𝑦 < π‘₯ ∨ 𝑦 = π‘₯)))
3736ancoms 460 . . . . . 6 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (𝑦 ≀ π‘₯ β†’ (𝑦 < π‘₯ ∨ 𝑦 = π‘₯)))
38373adant1 1131 . . . . 5 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (𝑦 ≀ π‘₯ β†’ (𝑦 < π‘₯ ∨ 𝑦 = π‘₯)))
39 simprl 770 . . . . . . . 8 ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) β†’ < Po 𝐡)
40 po2nr 5603 . . . . . . . . 9 (( < Po 𝐡 ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ Β¬ (π‘₯ < 𝑦 ∧ 𝑦 < π‘₯))
41403impb 1116 . . . . . . . 8 (( < Po 𝐡 ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ Β¬ (π‘₯ < 𝑦 ∧ 𝑦 < π‘₯))
4239, 41syl3an1 1164 . . . . . . 7 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ Β¬ (π‘₯ < 𝑦 ∧ 𝑦 < π‘₯))
4342pm2.21d 121 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ ((π‘₯ < 𝑦 ∧ 𝑦 < π‘₯) β†’ π‘₯ = 𝑦))
44 simpl 484 . . . . . . 7 ((π‘₯ = 𝑦 ∧ 𝑦 < π‘₯) β†’ π‘₯ = 𝑦)
4544a1i 11 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ ((π‘₯ = 𝑦 ∧ 𝑦 < π‘₯) β†’ π‘₯ = 𝑦))
46 simpr 486 . . . . . . . 8 ((π‘₯ < 𝑦 ∧ 𝑦 = π‘₯) β†’ 𝑦 = π‘₯)
4746equcomd 2023 . . . . . . 7 ((π‘₯ < 𝑦 ∧ 𝑦 = π‘₯) β†’ π‘₯ = 𝑦)
4847a1i 11 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ ((π‘₯ < 𝑦 ∧ 𝑦 = π‘₯) β†’ π‘₯ = 𝑦))
49 simpl 484 . . . . . . 7 ((π‘₯ = 𝑦 ∧ 𝑦 = π‘₯) β†’ π‘₯ = 𝑦)
5049a1i 11 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ ((π‘₯ = 𝑦 ∧ 𝑦 = π‘₯) β†’ π‘₯ = 𝑦))
5143, 45, 48, 50ccased 1038 . . . . 5 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (((π‘₯ < 𝑦 ∨ π‘₯ = 𝑦) ∧ (𝑦 < π‘₯ ∨ 𝑦 = π‘₯)) β†’ π‘₯ = 𝑦))
5235, 38, 51syl2and 609 . . . 4 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ π‘₯) β†’ π‘₯ = 𝑦))
53 simpr1 1195 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ π‘₯ ∈ 𝐡)
54 simpr2 1196 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ 𝑦 ∈ 𝐡)
5553, 54, 34syl2anc 585 . . . . 5 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ (π‘₯ ≀ 𝑦 β†’ (π‘₯ < 𝑦 ∨ π‘₯ = 𝑦)))
56 simpr3 1197 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ 𝑧 ∈ 𝐡)
573, 13, 1pleval2i 18289 . . . . . 6 ((𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) β†’ (𝑦 ≀ 𝑧 β†’ (𝑦 < 𝑧 ∨ 𝑦 = 𝑧)))
5854, 56, 57syl2anc 585 . . . . 5 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ (𝑦 ≀ 𝑧 β†’ (𝑦 < 𝑧 ∨ 𝑦 = 𝑧)))
59 potr 5602 . . . . . . . 8 (( < Po 𝐡 ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ < 𝑦 ∧ 𝑦 < 𝑧) β†’ π‘₯ < 𝑧))
6039, 59sylan 581 . . . . . . 7 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ < 𝑦 ∧ 𝑦 < 𝑧) β†’ π‘₯ < 𝑧))
61 simpll 766 . . . . . . . 8 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ 𝐾 ∈ 𝑉)
6213, 1pltle 18286 . . . . . . . 8 ((𝐾 ∈ 𝑉 ∧ π‘₯ ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) β†’ (π‘₯ < 𝑧 β†’ π‘₯ ≀ 𝑧))
6361, 53, 56, 62syl3anc 1372 . . . . . . 7 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ (π‘₯ < 𝑧 β†’ π‘₯ ≀ 𝑧))
6460, 63syld 47 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ < 𝑦 ∧ 𝑦 < 𝑧) β†’ π‘₯ ≀ 𝑧))
65 breq1 5152 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (π‘₯ < 𝑧 ↔ 𝑦 < 𝑧))
6665biimpar 479 . . . . . . 7 ((π‘₯ = 𝑦 ∧ 𝑦 < 𝑧) β†’ π‘₯ < 𝑧)
6766, 63syl5 34 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ = 𝑦 ∧ 𝑦 < 𝑧) β†’ π‘₯ ≀ 𝑧))
68 breq2 5153 . . . . . . . 8 (𝑦 = 𝑧 β†’ (π‘₯ < 𝑦 ↔ π‘₯ < 𝑧))
6968biimpac 480 . . . . . . 7 ((π‘₯ < 𝑦 ∧ 𝑦 = 𝑧) β†’ π‘₯ < 𝑧)
7069, 63syl5 34 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ < 𝑦 ∧ 𝑦 = 𝑧) β†’ π‘₯ ≀ 𝑧))
7153, 33syldan 592 . . . . . . 7 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ π‘₯ ≀ π‘₯)
72 eqtr 2756 . . . . . . . 8 ((π‘₯ = 𝑦 ∧ 𝑦 = 𝑧) β†’ π‘₯ = 𝑧)
7372breq2d 5161 . . . . . . 7 ((π‘₯ = 𝑦 ∧ 𝑦 = 𝑧) β†’ (π‘₯ ≀ π‘₯ ↔ π‘₯ ≀ 𝑧))
7471, 73syl5ibcom 244 . . . . . 6 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ = 𝑦 ∧ 𝑦 = 𝑧) β†’ π‘₯ ≀ 𝑧))
7564, 67, 70, 74ccased 1038 . . . . 5 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ (((π‘₯ < 𝑦 ∨ π‘₯ = 𝑦) ∧ (𝑦 < 𝑧 ∨ 𝑦 = 𝑧)) β†’ π‘₯ ≀ 𝑧))
7655, 58, 75syl2and 609 . . . 4 (((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))
7723, 24, 25, 33, 52, 76isposd 18276 . . 3 ((𝐾 ∈ 𝑉 ∧ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )) β†’ 𝐾 ∈ Poset)
7877ex 414 . 2 (𝐾 ∈ 𝑉 β†’ (( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ ) β†’ 𝐾 ∈ Poset))
7922, 78impbid2 225 1 (𝐾 ∈ 𝑉 β†’ (𝐾 ∈ Poset ↔ ( < Po 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  βŸ¨cop 4635   class class class wbr 5149  {copab 5211   I cid 5574   Po wpo 5587   β†Ύ cres 5679  Rel wrel 5682  β€˜cfv 6544  Basecbs 17144  lecple 17204  Posetcpo 18260  ltcplt 18261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-proset 18248  df-poset 18266  df-plt 18283
This theorem is referenced by:  tosso  18372
  Copyright terms: Public domain W3C validator