Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oawordex2 Structured version   Visualization version   GIF version

Theorem oawordex2 42066
Description: If 𝐶 is between 𝐴 (inclusive) and (𝐴 +o 𝐵) (exclusive), there is an ordinal which equals 𝐶 when summed to 𝐴. This is a slightly different statement than oawordex 8556 or oawordeu 8554. (Contributed by RP, 7-Jan-2025.)
Assertion
Ref Expression
oawordex2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥𝐵 (𝐴 +o 𝑥) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem oawordex2
StepHypRef Expression
1 simprl 769 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → 𝐴𝐶)
2 simpll 765 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → 𝐴 ∈ On)
3 oacl 8534 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4 simpr 485 . . . . 5 ((𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵)) → 𝐶 ∈ (𝐴 +o 𝐵))
5 onelon 6389 . . . . 5 (((𝐴 +o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 +o 𝐵)) → 𝐶 ∈ On)
63, 4, 5syl2an 596 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → 𝐶 ∈ On)
7 oawordex 8556 . . . 4 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐶 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶))
82, 6, 7syl2anc 584 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → (𝐴𝐶 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶))
91, 8mpbid 231 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶)
10 simprr 771 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝐴 +o 𝑥) = 𝐶)
11 simprr 771 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → 𝐶 ∈ (𝐴 +o 𝐵))
1211adantr 481 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐶 ∈ (𝐴 +o 𝐵))
1310, 12eqeltrd 2833 . . 3 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))
14 simprl 769 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝑥 ∈ On)
15 simpllr 774 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐵 ∈ On)
162adantr 481 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐴 ∈ On)
17 oaord 8546 . . . 4 ((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
1814, 15, 16, 17syl3anc 1371 . . 3 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
1913, 18mpbird 256 . 2 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝑥𝐵)
209, 19, 10reximssdv 3172 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥𝐵 (𝐴 +o 𝑥) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3070  wss 3948  Oncon0 6364  (class class class)co 7408   +o coa 8462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-oadd 8469
This theorem is referenced by:  nnawordexg  42067  tfsconcatlem  42076  tfsconcatfv  42081  tfsconcatrn  42082  tfsconcatrev  42088  oaun3lem1  42114  oadif1  42120
  Copyright terms: Public domain W3C validator