Theorem oasubex 41969

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 1138	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ On)
2		simp1 1137	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ On)
3		simp3 1139	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
4		oawordex 8553	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴))
5	4	biimpa 478	. . 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 486	. . . . . . 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 8549	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴))
11	8, 9, 10	syl2anc 585	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴))
12	11	adantr 482	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → 𝐴 ⊆ (𝐵 +o 𝐴))
13	7, 12	eqsstrd 4019	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴))
14		simpr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝑐 ∈ On)
15		oaword 8545	. . . . . . . 8 ⊢ ((𝑐 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑐 ⊆ 𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴)))
16	14, 8, 9, 15	syl3anc 1372	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → (𝑐 ⊆ 𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴)))
17	16	adantr 482	. . . . . 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 414	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → ((𝐵 +o 𝑐) = 𝐴 → 𝑐 ⊆ 𝐴))
20	19	ancrd 553	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → ((𝐵 +o 𝑐) = 𝐴 → (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴)))
21	20	reximdva 3169	. 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 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∃wrex 3071   ⊆ wss 3947  Oncon0 6361  (class class class)co 7404   +_o coa 8458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-oadd 8465

This theorem is referenced by: (None)