Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oasubex Structured version   Visualization version   GIF version

Theorem oasubex 42502
Description: While subtraction can't be a binary operation on ordinals, for any pair of ordinals there exists an ordinal that can be added to the lessor (or equal) one which will sum to the greater. Theorem 2.19 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.)
Assertion
Ref Expression
oasubex ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) → ∃𝑐 ∈ On (𝑐𝐴 ∧ (𝐵 +o 𝑐) = 𝐴))
Distinct variable groups:   𝐴,𝑐   𝐵,𝑐

Proof of Theorem oasubex
StepHypRef Expression
1 simp2 1136 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
2 simp1 1135 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) → 𝐴 ∈ On)
3 simp3 1137 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) → 𝐵𝐴)
4 oawordex 8563 . . . 4 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴))
54biimpa 476 . . 3 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵𝐴) → ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴)
61, 2, 3, 5syl21anc 835 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) → ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴)
7 simpr 484 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝐵 +o 𝑐) = 𝐴)
8 simpl1 1190 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) → 𝐴 ∈ On)
9 simpl2 1191 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) → 𝐵 ∈ On)
10 oaword2 8559 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴))
118, 9, 10syl2anc 583 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴))
1211adantr 480 . . . . . . 7 ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → 𝐴 ⊆ (𝐵 +o 𝐴))
137, 12eqsstrd 4020 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴))
14 simpr 484 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) → 𝑐 ∈ On)
15 oaword 8555 . . . . . . . 8 ((𝑐 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑐𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴)))
1614, 8, 9, 15syl3anc 1370 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) → (𝑐𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴)))
1716adantr 480 . . . . . 6 ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝑐𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴)))
1813, 17mpbird 257 . . . . 5 ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → 𝑐𝐴)
1918ex 412 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) → ((𝐵 +o 𝑐) = 𝐴𝑐𝐴))
2019ancrd 551 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) ∧ 𝑐 ∈ On) → ((𝐵 +o 𝑐) = 𝐴 → (𝑐𝐴 ∧ (𝐵 +o 𝑐) = 𝐴)))
2120reximdva 3167 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) → (∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴 → ∃𝑐 ∈ On (𝑐𝐴 ∧ (𝐵 +o 𝑐) = 𝐴)))
226, 21mpd 15 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵𝐴) → ∃𝑐 ∈ On (𝑐𝐴 ∧ (𝐵 +o 𝑐) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wrex 3069  wss 3948  Oncon0 6364  (class class class)co 7412   +o coa 8469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-oadd 8476
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator