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Theorem oawordex2 42820
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 8576 or oawordeu 8574. (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 8554 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4 simpr 483 . . . . 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 594 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → 𝐶 ∈ On)
7 oawordex 8576 . . . 4 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐶 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶))
82, 6, 7syl2anc 582 . . 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 479 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐶 ∈ (𝐴 +o 𝐵))
1310, 12eqeltrd 2825 . . 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 479 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐴 ∈ On)
17 oaord 8566 . . . 4 ((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
1814, 15, 16, 17syl3anc 1368 . . 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 3163 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 394   = wceq 1533  wcel 2098  wrex 3060  wss 3939  Oncon0 6364  (class class class)co 7416   +o coa 8482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-oadd 8489
This theorem is referenced by:  nnawordexg  42821  tfsconcatlem  42830  tfsconcatfv  42835  tfsconcatrn  42836  tfsconcatrev  42842  oaun3lem1  42868  oadif1  42874
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