Theorem oawordex2 43308

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 8498 or oawordeu 8496. (Contributed by RP, 7-Jan-2025.)

Assertion

Ref	Expression
oawordex2	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem oawordex2

Step	Hyp	Ref	Expression
1		simprl 770	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → 𝐴 ⊆ 𝐶)
2		simpll 766	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → 𝐴 ∈ On)
3		oacl 8476	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4		simpr 484	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵)) → 𝐶 ∈ (𝐴 +o 𝐵))
5		onelon 6345	. . . . 5 ⊢ (((𝐴 +o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 +o 𝐵)) → 𝐶 ∈ On)
6	3, 4, 5	syl2an 596	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → 𝐶 ∈ On)
7		oawordex 8498	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐶 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶))
8	2, 6, 7	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → (𝐴 ⊆ 𝐶 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶))
9	1, 8	mpbid 232	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶)
10		simprr 772	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝐴 +o 𝑥) = 𝐶)
11		simprr 772	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → 𝐶 ∈ (𝐴 +o 𝐵))
12	11	adantr 480	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐶 ∈ (𝐴 +o 𝐵))
13	10, 12	eqeltrd 2828	. . 3 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))
14		simprl 770	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝑥 ∈ On)
15		simpllr 775	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐵 ∈ On)
16	2	adantr 480	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐴 ∈ On)
17		oaord 8488	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
18	14, 15, 16, 17	syl3anc 1373	. . 3 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
19	13, 18	mpbird 257	. 2 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝑥 ∈ 𝐵)
20	9, 19, 10	reximssdv 3151	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3911  Oncon0 6320  (class class class)co 7369   +_o coa 8408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-oadd 8415

This theorem is referenced by:  nnawordexg  43309  tfsconcatlem  43318  tfsconcatfv  43323  tfsconcatrn  43324  tfsconcatrev  43330  oaun3lem1  43356  oadif1  43362