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Mirrors > Home > MPE Home > Th. List > axgroth2 | Structured version Visualization version GIF version |
Description: Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.) |
Ref | Expression |
---|---|
axgroth2 | ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-groth 10818 | . 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))) | |
2 | ssdomg 8996 | . . . . . . . . . 10 ⊢ (𝑦 ∈ V → (𝑧 ⊆ 𝑦 → 𝑧 ≼ 𝑦)) | |
3 | 2 | elv 3481 | . . . . . . . . 9 ⊢ (𝑧 ⊆ 𝑦 → 𝑧 ≼ 𝑦) |
4 | 3 | biantrurd 534 | . . . . . . . 8 ⊢ (𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ↔ (𝑧 ≼ 𝑦 ∧ 𝑦 ≼ 𝑧))) |
5 | sbthb 9094 | . . . . . . . 8 ⊢ ((𝑧 ≼ 𝑦 ∧ 𝑦 ≼ 𝑧) ↔ 𝑧 ≈ 𝑦) | |
6 | 4, 5 | bitrdi 287 | . . . . . . 7 ⊢ (𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ↔ 𝑧 ≈ 𝑦)) |
7 | 6 | orbi1d 916 | . . . . . 6 ⊢ (𝑧 ⊆ 𝑦 → ((𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))) |
8 | 7 | pm5.74i 271 | . . . . 5 ⊢ ((𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)) ↔ (𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))) |
9 | 8 | albii 1822 | . . . 4 ⊢ (∀𝑧(𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)) ↔ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))) |
10 | 9 | 3anbi3i 1160 | . . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))) |
11 | 10 | exbii 1851 | . 2 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))) |
12 | 1, 11 | mpbir 230 | 1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 ∀wal 1540 ∃wex 1782 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⊆ wss 3949 class class class wbr 5149 ≈ cen 8936 ≼ cdom 8937 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-groth 10818 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-er 8703 df-en 8940 df-dom 8941 |
This theorem is referenced by: axgroth3 10826 |
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