Theorem oasubex 43393

Description: While subtraction can't be a binary operation on ordinals, for any pair of ordinals there exists an ordinal that can be added to the lessor (or equal) one which will sum to the greater. Theorem 2.19 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.)

Assertion

Ref	Expression
oasubex	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴))

Distinct variable groups:   𝐴,𝑐   𝐵,𝑐

Proof of Theorem oasubex

Step	Hyp	Ref	Expression
1		simp2 1137	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ On)
2		simp1 1136	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ On)
3		simp3 1138	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
4		oawordex 8481	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴))
5	4	biimpa 476	. . 3 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴)
6	1, 2, 3, 5	syl21anc 837	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴)
7		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝐵 +o 𝑐) = 𝐴)
8		simpl1 1192	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝐴 ∈ On)
9		simpl2 1193	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝐵 ∈ On)
10		oaword2 8477	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴))
11	8, 9, 10	syl2anc 584	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴))
12	11	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → 𝐴 ⊆ (𝐵 +o 𝐴))
13	7, 12	eqsstrd 3966	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴))
14		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝑐 ∈ On)
15		oaword 8473	. . . . . . . 8 ⊢ ((𝑐 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑐 ⊆ 𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴)))
16	14, 8, 9, 15	syl3anc 1373	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → (𝑐 ⊆ 𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴)))
17	16	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝑐 ⊆ 𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴)))
18	13, 17	mpbird 257	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → 𝑐 ⊆ 𝐴)
19	18	ex 412	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → ((𝐵 +o 𝑐) = 𝐴 → 𝑐 ⊆ 𝐴))
20	19	ancrd 551	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → ((𝐵 +o 𝑐) = 𝐴 → (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴)))
21	20	reximdva 3147	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → (∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴 → ∃𝑐 ∈ On (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴)))
22	6, 21	mpd 15	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3058   ⊆ wss 3899  Oncon0 6314  (class class class)co 7355   +_o coa 8391

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-oadd 8398

This theorem is referenced by: (None)