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Mirrors > Home > MPE Home > Th. List > en1eqsnOLD | Structured version Visualization version GIF version |
Description: Obsolete version of en1eqsn 9276 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 8641 | . . . . . 6 ⊢ 1o ∈ ω | |
2 | ssid 3999 | . . . . . 6 ⊢ 1o ⊆ 1o | |
3 | ssnnfi 9171 | . . . . . 6 ⊢ ((1o ∈ ω ∧ 1o ⊆ 1o) → 1o ∈ Fin) | |
4 | 1, 2, 3 | mp2an 689 | . . . . 5 ⊢ 1o ∈ Fin |
5 | enfii 9191 | . . . . 5 ⊢ ((1o ∈ Fin ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) | |
6 | 4, 5 | mpan 687 | . . . 4 ⊢ (𝐵 ≈ 1o → 𝐵 ∈ Fin) |
7 | 6 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) |
8 | snssi 4806 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ⊆ 𝐵) |
10 | ensn1g 9021 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o) | |
11 | ensym 9001 | . . . 4 ⊢ (𝐵 ≈ 1o → 1o ≈ 𝐵) | |
12 | entr 9004 | . . . 4 ⊢ (({𝐴} ≈ 1o ∧ 1o ≈ 𝐵) → {𝐴} ≈ 𝐵) | |
13 | 10, 11, 12 | syl2an 595 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ≈ 𝐵) |
14 | fisseneq 9259 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ {𝐴} ⊆ 𝐵 ∧ {𝐴} ≈ 𝐵) → {𝐴} = 𝐵) | |
15 | 7, 9, 13, 14 | syl3anc 1368 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} = 𝐵) |
16 | 15 | eqcomd 2732 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 {csn 4623 class class class wbr 5141 ωcom 7852 1oc1o 8460 ≈ cen 8938 Fincfn 8941 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 |
This theorem is referenced by: (None) |
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