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Mirrors > Home > MPE Home > Th. List > numthcor | Structured version Visualization version GIF version |
Description: Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
numthcor | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5109 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ 𝑥 ↔ 𝐴 ≺ 𝑥)) | |
2 | 1 | rexbidv 3176 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ≺ 𝑥 ↔ ∃𝑥 ∈ On 𝐴 ≺ 𝑥)) |
3 | vpwex 5333 | . . . 4 ⊢ 𝒫 𝑦 ∈ V | |
4 | 3 | numth2 10408 | . . 3 ⊢ ∃𝑥 ∈ On 𝑥 ≈ 𝒫 𝑦 |
5 | vex 3450 | . . . . . 6 ⊢ 𝑦 ∈ V | |
6 | 5 | canth2 9075 | . . . . 5 ⊢ 𝑦 ≺ 𝒫 𝑦 |
7 | ensym 8944 | . . . . 5 ⊢ (𝑥 ≈ 𝒫 𝑦 → 𝒫 𝑦 ≈ 𝑥) | |
8 | sdomentr 9056 | . . . . 5 ⊢ ((𝑦 ≺ 𝒫 𝑦 ∧ 𝒫 𝑦 ≈ 𝑥) → 𝑦 ≺ 𝑥) | |
9 | 6, 7, 8 | sylancr 588 | . . . 4 ⊢ (𝑥 ≈ 𝒫 𝑦 → 𝑦 ≺ 𝑥) |
10 | 9 | reximi 3088 | . . 3 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝒫 𝑦 → ∃𝑥 ∈ On 𝑦 ≺ 𝑥) |
11 | 4, 10 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ On 𝑦 ≺ 𝑥 |
12 | 2, 11 | vtoclg 3526 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∃wrex 3074 𝒫 cpw 4561 class class class wbr 5106 Oncon0 6318 ≈ cen 8881 ≺ csdm 8883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-ac2 10400 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-card 9876 df-ac 10053 |
This theorem is referenced by: cardmin 10501 |
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