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Mirrors > Home > MPE Home > Th. List > Mathboxes > oenassex | Structured version Visualization version GIF version |
Description: Ordinal two raised to two to the zeroth power is not the same as two squared then raised to the zeroth power. (Contributed by RP, 30-Jan-2025.) |
Ref | Expression |
---|---|
oenassex | ⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8460 | . . . 4 ⊢ 1o ∈ V | |
2 | 1 | prid2 4761 | . . 3 ⊢ 1o ∈ {∅, 1o} |
3 | df2o3 8458 | . . 3 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2832 | . 2 ⊢ 1o ∈ 2o |
5 | elneq 9577 | . . 3 ⊢ (1o ∈ 2o → 1o ≠ 2o) | |
6 | df-ne 2941 | . . . 4 ⊢ (2o ≠ 1o ↔ ¬ 2o = 1o) | |
7 | necom 2994 | . . . 4 ⊢ (1o ≠ 2o ↔ 2o ≠ 1o) | |
8 | 2on 8464 | . . . . . . . . 9 ⊢ 2o ∈ On | |
9 | oe0 8506 | . . . . . . . . 9 ⊢ (2o ∈ On → (2o ↑o ∅) = 1o) | |
10 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ (2o ↑o ∅) = 1o |
11 | 10 | oveq2i 7405 | . . . . . . 7 ⊢ (2o ↑o (2o ↑o ∅)) = (2o ↑o 1o) |
12 | oe1 8529 | . . . . . . . 8 ⊢ (2o ∈ On → (2o ↑o 1o) = 2o) | |
13 | 8, 12 | ax-mp 5 | . . . . . . 7 ⊢ (2o ↑o 1o) = 2o |
14 | 11, 13 | eqtri 2760 | . . . . . 6 ⊢ (2o ↑o (2o ↑o ∅)) = 2o |
15 | 8, 8 | pm3.2i 471 | . . . . . . 7 ⊢ (2o ∈ On ∧ 2o ∈ On) |
16 | oecl 8521 | . . . . . . 7 ⊢ ((2o ∈ On ∧ 2o ∈ On) → (2o ↑o 2o) ∈ On) | |
17 | oe0 8506 | . . . . . . 7 ⊢ ((2o ↑o 2o) ∈ On → ((2o ↑o 2o) ↑o ∅) = 1o) | |
18 | 15, 16, 17 | mp2b 10 | . . . . . 6 ⊢ ((2o ↑o 2o) ↑o ∅) = 1o |
19 | 14, 18 | eqeq12i 2750 | . . . . 5 ⊢ ((2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) ↔ 2o = 1o) |
20 | 19 | notbii 319 | . . . 4 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) ↔ ¬ 2o = 1o) |
21 | 6, 7, 20 | 3bitr4i 302 | . . 3 ⊢ (1o ≠ 2o ↔ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)) |
22 | 5, 21 | sylib 217 | . 2 ⊢ (1o ∈ 2o → ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)) |
23 | 4, 22 | ax-mp 5 | 1 ⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∅c0 4319 {cpr 4625 Oncon0 6354 (class class class)co 7394 1oc1o 8443 2oc2o 8444 ↑o coe 8449 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 ax-reg 9571 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-2o 8451 df-oadd 8454 df-omul 8455 df-oexp 8456 |
This theorem is referenced by: oenass 41904 |
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