Theorem oawordex2 43338

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 8467 or oawordeu 8465. (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 8445	. . . . 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 6327	. . . . 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 8467	. . . 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 2829	. . 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 8457	. . . 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 3148	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 1541   ∈ wcel 2110  ∃wrex 3054   ⊆ wss 3900  Oncon0 6302  (class class class)co 7341   +_o coa 8377

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 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-oadd 8384

This theorem is referenced by:  nnawordexg  43339  tfsconcatlem  43348  tfsconcatfv  43353  tfsconcatrn  43354  tfsconcatrev  43360  oaun3lem1  43386  oadif1  43392