Theorem oawordex2 44271

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 8543 or oawordeu 8541. (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 783	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → 𝐴 ⊆ 𝐶)
2		simpll 779	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → 𝐴 ∈ On)
3		oacl 8521	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
4		simpr 490	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵)) → 𝐶 ∈ (𝐴 +o 𝐵))
5		onelon 6376	. . . . 5 ⊢ (((𝐴 +o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 +o 𝐵)) → 𝐶 ∈ On)
6	3, 4, 5	syl2an 608	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → 𝐶 ∈ On)
7		oawordex 8543	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐶 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶))
8	2, 6, 7	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → (𝐴 ⊆ 𝐶 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶))
9	1, 8	mpbid 235	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐶)
10		simprr 785	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → (𝐴 +o 𝑥) = 𝐶)
11		simprr 785	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → 𝐶 ∈ (𝐴 +o 𝐵))
12	11	adantr 486	. . . 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 2860	. . 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 783	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝑥 ∈ On)
15		simpllr 788	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐵 ∈ On)
16	2	adantr 486	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝐴 ∈ On)
17		oaord 8533	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
18	14, 15, 16, 17	syl3anc 1398	. . 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 260	. 2 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) ∧ (𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐶)) → 𝑥 ∈ 𝐵)
20	9, 19, 10	reximssdv 3180	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3898  Oncon0 6351  (class class class)co 7408   +_o coa 8451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8458

This theorem is used by:  nnawordexg  44272  tfsconcatlem  44281  tfsconcatfv  44286  tfsconcatrn  44287  tfsconcatrev  44293  oaun3lem1  44319  oadif1  44325