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Mirrors > Home > MPE Home > Th. List > ssnnfiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ssnnfi 8825 as of 23-Sep-2024. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ssnnfiOLD | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspss 4000 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴)) | |
2 | pssnn 8824 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥) | |
3 | elnn 7633 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑥 ∈ ω) | |
4 | 3 | expcom 417 | . . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑥 ∈ 𝐴 → 𝑥 ∈ ω)) |
5 | 4 | anim1d 614 | . . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ 𝐴 ∧ 𝐵 ≈ 𝑥) → (𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥))) |
6 | 5 | reximdv2 3180 | . . . . . 6 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥 → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)) |
7 | 6 | adantr 484 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → (∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥 → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)) |
8 | 2, 7 | mpd 15 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) |
9 | eleq1 2818 | . . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵 ∈ ω ↔ 𝐴 ∈ ω)) | |
10 | 9 | biimparc 483 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐵 ∈ ω) |
11 | enrefnn 8702 | . . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ≈ 𝐵) | |
12 | breq2 5043 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐵 ≈ 𝑥 ↔ 𝐵 ≈ 𝐵)) | |
13 | 12 | rspcev 3527 | . . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐵 ≈ 𝐵) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) |
14 | 10, 11, 13 | syl2anc2 588 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) |
15 | 8, 14 | jaodan 958 | . . 3 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴)) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) |
16 | 1, 15 | sylan2b 597 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) |
17 | isfi 8630 | . 2 ⊢ (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) | |
18 | 16, 17 | sylibr 237 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 ⊆ wss 3853 ⊊ wpss 3854 class class class wbr 5039 ωcom 7622 ≈ cen 8601 Fincfn 8604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-om 7623 df-en 8605 df-fin 8608 |
This theorem is referenced by: (None) |
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