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Mirrors > Home > MPE Home > Th. List > en1eqsnOLD | Structured version Visualization version GIF version |
Description: Obsolete version of en1eqsn 9270 as of 4-Jan-2025. (Contributed by FL, 18-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
en1eqsnOLD | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8635 | . . . . . 6 ⊢ 1o ∈ ω | |
2 | ssid 4003 | . . . . . 6 ⊢ 1o ⊆ 1o | |
3 | ssnnfi 9165 | . . . . . 6 ⊢ ((1o ∈ ω ∧ 1o ⊆ 1o) → 1o ∈ Fin) | |
4 | 1, 2, 3 | mp2an 690 | . . . . 5 ⊢ 1o ∈ Fin |
5 | enfii 9185 | . . . . 5 ⊢ ((1o ∈ Fin ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) | |
6 | 4, 5 | mpan 688 | . . . 4 ⊢ (𝐵 ≈ 1o → 𝐵 ∈ Fin) |
7 | 6 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) |
8 | snssi 4810 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ⊆ 𝐵) |
10 | ensn1g 9015 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o) | |
11 | ensym 8995 | . . . 4 ⊢ (𝐵 ≈ 1o → 1o ≈ 𝐵) | |
12 | entr 8998 | . . . 4 ⊢ (({𝐴} ≈ 1o ∧ 1o ≈ 𝐵) → {𝐴} ≈ 𝐵) | |
13 | 10, 11, 12 | syl2an 596 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ≈ 𝐵) |
14 | fisseneq 9253 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ {𝐴} ⊆ 𝐵 ∧ {𝐴} ≈ 𝐵) → {𝐴} = 𝐵) | |
15 | 7, 9, 13, 14 | syl3anc 1371 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} = 𝐵) |
16 | 15 | eqcomd 2738 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 {csn 4627 class class class wbr 5147 ωcom 7851 1oc1o 8455 ≈ cen 8932 Fincfn 8935 |
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-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7852 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 |
This theorem is referenced by: (None) |
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