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Mirrors > Home > MPE Home > Th. List > exel | Structured version Visualization version GIF version |
Description: There exist two sets, one
a member of the other.
This theorem looks similar to el 5437, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1811, ax-6 1971, and ax-pr 5427. This theorem does not exclude that these two sets could actually be one single set containing itself. That two different sets exist is proved by exexneq 5434. (Contributed by SN, 23-Dec-2024.) |
Ref | Expression |
---|---|
exel | ⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pr 5427 | . 2 ⊢ ∃𝑦∀𝑥((𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → 𝑥 ∈ 𝑦) | |
2 | ax6ev 1973 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑧 | |
3 | pm2.07 901 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑧 ∨ 𝑥 = 𝑧)) | |
4 | 2, 3 | eximii 1839 | . . 3 ⊢ ∃𝑥(𝑥 = 𝑧 ∨ 𝑥 = 𝑧) |
5 | exim 1836 | . . 3 ⊢ (∀𝑥((𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → 𝑥 ∈ 𝑦) → (∃𝑥(𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → ∃𝑥 𝑥 ∈ 𝑦)) | |
6 | 4, 5 | mpi 20 | . 2 ⊢ (∀𝑥((𝑥 = 𝑧 ∨ 𝑥 = 𝑧) → 𝑥 ∈ 𝑦) → ∃𝑥 𝑥 ∈ 𝑦) |
7 | 1, 6 | eximii 1839 | 1 ⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 ∀wal 1539 ∃wex 1781 |
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-6 1971 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-or 846 df-ex 1782 |
This theorem is referenced by: exexneq 5434 |
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