Theorem oenass 43735

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 43734	. 2 ⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)
2		2on 8407	. . 3 ⊢ 2o ∈ On
3		0elon 6367	. . . . 5 ⊢ ∅ ∈ On
4		oveq2 7364	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (2o ↑o 𝑐) = (2o ↑o ∅))
5	4	oveq2d 7372	. . . . . . . 8 ⊢ (𝑐 = ∅ → (2o ↑o (2o ↑o 𝑐)) = (2o ↑o (2o ↑o ∅)))
6		oveq2 7364	. . . . . . . 8 ⊢ (𝑐 = ∅ → ((2o ↑o 2o) ↑o 𝑐) = ((2o ↑o 2o) ↑o ∅))
7	5, 6	eqeq12d 2751	. . . . . . 7 ⊢ (𝑐 = ∅ → ((2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐) ↔ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)))
8	7	notbid 318	. . . . . 6 ⊢ (𝑐 = ∅ → (¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐) ↔ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)))
9	8	rspcev 3562	. . . . 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 691	. . . 4 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐))
11		oveq1 7363	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ↑o 𝑐) = (2o ↑o 𝑐))
12	11	oveq2d 7372	. . . . . . . 8 ⊢ (𝑏 = 2o → (2o ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (2o ↑o 𝑐)))
13		oveq2 7364	. . . . . . . . 9 ⊢ (𝑏 = 2o → (2o ↑o 𝑏) = (2o ↑o 2o))
14	13	oveq1d 7371	. . . . . . . 8 ⊢ (𝑏 = 2o → ((2o ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 2o) ↑o 𝑐))
15	12, 14	eqeq12d 2751	. . . . . . 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 318	. . . . . 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 3159	. . . . 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 3562	. . . 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 588	. . 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 7363	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (𝑏 ↑o 𝑐)))
21		oveq1 7363	. . . . . . . . 9 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o ↑o 𝑏))
22	21	oveq1d 7371	. . . . . . . 8 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 𝑏) ↑o 𝑐))
23	20, 22	eqeq12d 2751	. . . . . . 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 318	. . . . . 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 3159	. . . . 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 3159	. . . 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 3562	. . 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 588	. 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)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  ∃wrex 3059  ∅c0 4263  Oncon0 6312  (class class class)co 7356  2_oc2o 8388   ↑_o coe 8393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678  ax-reg 9496

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  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 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-oadd 8398  df-omul 8399  df-oexp 8400

This theorem is referenced by: (None)