Theorem oasubex 43269

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 8475	. . . 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 8471	. . . . . . . . 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 3970	. . . . . 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 8467	. . . . . . . 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 3142	. 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 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3903  Oncon0 6307  (class class class)co 7349   +_o coa 8385

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 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  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 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-oadd 8392

This theorem is referenced by: (None)