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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 8471 | . . . 4 ⊢ 1o ∈ V | |
2 | 1 | prid2 4759 | . . 3 ⊢ 1o ∈ {∅, 1o} |
3 | df2o3 8469 | . . 3 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2824 | . 2 ⊢ 1o ∈ 2o |
5 | elneq 9588 | . . 3 ⊢ (1o ∈ 2o → 1o ≠ 2o) | |
6 | df-ne 2933 | . . . 4 ⊢ (2o ≠ 1o ↔ ¬ 2o = 1o) | |
7 | necom 2986 | . . . 4 ⊢ (1o ≠ 2o ↔ 2o ≠ 1o) | |
8 | 2on 8475 | . . . . . . . . 9 ⊢ 2o ∈ On | |
9 | oe0 8517 | . . . . . . . . 9 ⊢ (2o ∈ On → (2o ↑o ∅) = 1o) | |
10 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ (2o ↑o ∅) = 1o |
11 | 10 | oveq2i 7412 | . . . . . . 7 ⊢ (2o ↑o (2o ↑o ∅)) = (2o ↑o 1o) |
12 | oe1 8539 | . . . . . . . 8 ⊢ (2o ∈ On → (2o ↑o 1o) = 2o) | |
13 | 8, 12 | ax-mp 5 | . . . . . . 7 ⊢ (2o ↑o 1o) = 2o |
14 | 11, 13 | eqtri 2752 | . . . . . 6 ⊢ (2o ↑o (2o ↑o ∅)) = 2o |
15 | 8, 8 | pm3.2i 470 | . . . . . . 7 ⊢ (2o ∈ On ∧ 2o ∈ On) |
16 | oecl 8532 | . . . . . . 7 ⊢ ((2o ∈ On ∧ 2o ∈ On) → (2o ↑o 2o) ∈ On) | |
17 | oe0 8517 | . . . . . . 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 2742 | . . . . 5 ⊢ ((2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) ↔ 2o = 1o) |
20 | 19 | notbii 320 | . . . 4 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) ↔ ¬ 2o = 1o) |
21 | 6, 7, 20 | 3bitr4i 303 | . . 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 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∅c0 4314 {cpr 4622 Oncon0 6354 (class class class)co 7401 1oc1o 8454 2oc2o 8455 ↑o coe 8460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 ax-reg 9582 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 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 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-oexp 8467 |
This theorem is referenced by: oenass 42524 |
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