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Mirrors > Home > MPE Home > Th. List > snnen2o | Structured version Visualization version GIF version |
Description: A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.) Avoid ax-pow 5361, ax-un 7738. (Revised by BTernaryTau, 1-Dec-2024.) |
Ref | Expression |
---|---|
snnen2o | ⊢ ¬ {𝐴} ≈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df2o3 8496 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
2 | 0ex 5304 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
3 | 1oex 8498 | . . . . . . . . 9 ⊢ 1o ∈ V | |
4 | 1n0 8510 | . . . . . . . . . 10 ⊢ 1o ≠ ∅ | |
5 | 4 | necomi 2985 | . . . . . . . . 9 ⊢ ∅ ≠ 1o |
6 | prnesn 4858 | . . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥}) | |
7 | 2, 3, 5, 6 | mp3an 1458 | . . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥} |
8 | 1, 7 | eqnetri 3001 | . . . . . . 7 ⊢ 2o ≠ {𝑥} |
9 | 8 | neii 2932 | . . . . . 6 ⊢ ¬ 2o = {𝑥} |
10 | 9 | nex 1795 | . . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥} |
11 | 2on0 8504 | . . . . . 6 ⊢ 2o ≠ ∅ | |
12 | f1cdmsn 7288 | . . . . . 6 ⊢ ((◡𝑓:2o–1-1→{𝐴} ∧ 2o ≠ ∅) → ∃𝑥2o = {𝑥}) | |
13 | 11, 12 | mpan2 689 | . . . . 5 ⊢ (◡𝑓:2o–1-1→{𝐴} → ∃𝑥2o = {𝑥}) |
14 | 10, 13 | mto 196 | . . . 4 ⊢ ¬ ◡𝑓:2o–1-1→{𝐴} |
15 | f1ocnv 6847 | . . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴}) | |
16 | f1of1 6834 | . . . . 5 ⊢ (◡𝑓:2o–1-1-onto→{𝐴} → ◡𝑓:2o–1-1→{𝐴}) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1→{𝐴}) |
18 | 14, 17 | mto 196 | . . 3 ⊢ ¬ 𝑓:{𝐴}–1-1-onto→2o |
19 | 18 | nex 1795 | . 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o |
20 | snex 5429 | . . 3 ⊢ {𝐴} ∈ V | |
21 | 2oex 8499 | . . 3 ⊢ 2o ∈ V | |
22 | breng 8975 | . . 3 ⊢ (({𝐴} ∈ V ∧ 2o ∈ V) → ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o)) | |
23 | 20, 21, 22 | mp2an 690 | . 2 ⊢ ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o) |
24 | 19, 23 | mtbir 322 | 1 ⊢ ¬ {𝐴} ≈ 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ∅c0 4322 {csn 4623 {cpr 4625 class class class wbr 5145 ◡ccnv 5673 –1-1→wf1 6543 –1-1-onto→wf1o 6545 1oc1o 8481 2oc2o 8482 ≈ cen 8963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-1o 8488 df-2o 8489 df-en 8967 |
This theorem is referenced by: 1sdom2 9267 1sdom2dom 9274 pr2ne 10040 pmtrsn 19513 trivnsimpgd 20093 |
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