Theorem oenass 44321

Description: Ordinal exponentiation is not associative. Remark 4.6 of [Schloeder] p. 14. (Contributed by RP, 30-Jan-2025.)

Assertion

Ref	Expression
oenass	⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐)

Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem oenass

Step	Hyp	Ref	Expression
1		oenassex 44320	. 2 ⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)
2		2on 8490	. . 3 ⊢ 2o ∈ On
3		0elon 6418	. . . . 5 ⊢ ∅ ∈ On
4		oveq2 7428	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (2o ↑o 𝑐) = (2o ↑o ∅))
5	4	oveq2d 7436	. . . . . . . 8 ⊢ (𝑐 = ∅ → (2o ↑o (2o ↑o 𝑐)) = (2o ↑o (2o ↑o ∅)))
6		oveq2 7428	. . . . . . . 8 ⊢ (𝑐 = ∅ → ((2o ↑o 2o) ↑o 𝑐) = ((2o ↑o 2o) ↑o ∅))
7	5, 6	eqeq12d 2776	. . . . . . 7 ⊢ (𝑐 = ∅ → ((2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐) ↔ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)))
8	7	notbid 321	. . . . . 6 ⊢ (𝑐 = ∅ → (¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐) ↔ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)))
9	8	rspcev 3576	. . . . 5 ⊢ ((∅ ∈ On ∧ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)) → ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐))
10	3, 9	mpan 703	. . . 4 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐))
11		oveq1 7427	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ↑o 𝑐) = (2o ↑o 𝑐))
12	11	oveq2d 7436	. . . . . . . 8 ⊢ (𝑏 = 2o → (2o ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (2o ↑o 𝑐)))
13		oveq2 7428	. . . . . . . . 9 ⊢ (𝑏 = 2o → (2o ↑o 𝑏) = (2o ↑o 2o))
14	13	oveq1d 7435	. . . . . . . 8 ⊢ (𝑏 = 2o → ((2o ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 2o) ↑o 𝑐))
15	12, 14	eqeq12d 2776	. . . . . . 7 ⊢ (𝑏 = 2o → ((2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐) ↔ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)))
16	15	notbid 321	. . . . . 6 ⊢ (𝑏 = 2o → (¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐) ↔ ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)))
17	16	rexbidv 3186	. . . . 5 ⊢ (𝑏 = 2o → (∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐) ↔ ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)))
18	17	rspcev 3576	. . . 4 ⊢ ((2o ∈ On ∧ ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐))
19	2, 10, 18	sylancr 599	. . 3 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐))
20		oveq1 7427	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (𝑏 ↑o 𝑐)))
21		oveq1 7427	. . . . . . . . 9 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o ↑o 𝑏))
22	21	oveq1d 7435	. . . . . . . 8 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 𝑏) ↑o 𝑐))
23	20, 22	eqeq12d 2776	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
24	23	notbid 321	. . . . . 6 ⊢ (𝑎 = 2o → (¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
25	24	rexbidv 3186	. . . . 5 ⊢ (𝑎 = 2o → (∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
26	25	rexbidv 3186	. . . 4 ⊢ (𝑎 = 2o → (∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
27	26	rspcev 3576	. . 3 ⊢ ((2o ∈ On ∧ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐))
28	2, 19, 27	sylancr 599	. 2 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐))
29	1, 28	ax-mp 5	1 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  ∅c0 4278  Oncon0 6362  (class class class)co 7420  2_oc2o 8470   ↑_o coe 8475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751  ax-reg 9586

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-omul 8481  df-oexp 8482

This theorem is used by: (None)