Theorem oawordex2 43424

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 8478 or oawordeu 8476. (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 8456	. . . . 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 6337	. . . . 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 8478	. . . 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 2831	. . 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 8468	. . . 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 3150	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 2111  ∃wrex 3056   ⊆ wss 3897  Oncon0 6312  (class class class)co 7352   +_o coa 8388

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-oadd 8395

This theorem is referenced by:  nnawordexg  43425  tfsconcatlem  43434  tfsconcatfv  43439  tfsconcatrn  43440  tfsconcatrev  43446  oaun3lem1  43472  oadif1  43478