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Theorem oawordex2 41704
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 8505 or oawordeu 8503. (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 770 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → 𝐴𝐶)
2 simpll 766 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → 𝐴 ∈ On)
3 oacl 8482 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4 simpr 486 . . . . 5 ((𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵)) → 𝐶 ∈ (𝐴 +o 𝐵))
5 onelon 6343 . . . . 5 (((𝐴 +o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 +o 𝐵)) → 𝐶 ∈ On)
63, 4, 5syl2an 597 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → 𝐶 ∈ On)
7 oawordex 8505 . . . 4 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐶 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶))
82, 6, 7syl2anc 585 . . 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 772 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝐴 +o 𝑥) = 𝐶)
11 simprr 772 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) → 𝐶 ∈ (𝐴 +o 𝐵))
1211adantr 482 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐶 ∈ (𝐴 +o 𝐵))
1310, 12eqeltrd 2834 . . 3 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))
14 simprl 770 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝑥 ∈ On)
15 simpllr 775 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐵 ∈ On)
162adantr 482 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐴 ∈ On)
17 oaord 8495 . . . 4 ((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
1814, 15, 16, 17syl3anc 1372 . . 3 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
1913, 18mpbird 257 . 2 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴𝐶𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝑥𝐵)
209, 19, 10reximssdv 3166 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 397   = wceq 1542  wcel 2107  wrex 3070  wss 3911  Oncon0 6318  (class class class)co 7358   +o coa 8410
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-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-oadd 8417
This theorem is referenced by:  nnawordexg  41705  oaun3lem1  41733  oadif1  41739
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